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Series of squares Prove that if gak is a convergent series

ISBN: 9780321947345 167

Solution for problem 78 Chapter 8.5

Calculus: Early Transcendentals | 2nd Edition

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Calculus: Early Transcendentals | 2nd Edition

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Problem 78

Series of squares Prove that if gak is a convergent series of positive terms, then the series gak 2 also converges.

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Exam 1 Review Sheet 1 Psych 380 Historical context—know explanations for abnormality, general approach to treatment, and overall evaluation for each approach to abnormal behavior. Definition of a psychological disorder  Psychological dysfunction ­ breakdown in cognitive, emotional, or behavioral functioning  Personal distress­ the person finds it upsetting  Atypical or not culturally expected Supernatural tradition  Explanation­ abnormal behavior caused by supernatural phenomenon (evil spirits/witches)  Treatment supernatural (exorcism, kill the witch)  Evaluation­ power of suggestion (placebo effect), but not in most cases; could stigmatize sufferers  Incidence­ number of new cases of a disorder appearing during a specific period  Prevalence­ number of people displaying a disorder in the total population at any given time  Prognosis­ predicted development and change of a disorder over time  Etiology­ cause or source of a disorder Trephination  (Neolithic societies 5000 BCE) removal of a piece of the skill, possibly to relieve pressure after injury Biological tradition  Hippocrates o Theorized physiological processes like intelligence, consciousness, and emotions originate in the brain o Determined stress affects psychological functioning  4 humors o Blood­(from heart)­excess creates cheerfulness (sanguinity) o Black bile (from spleen)­ produces depression (melancholy) o Yellow bile (from liver)­ produces anger (choler) o Phlegm (from brain)­ produces lethargy  General paresis­ progressive mental illness involving delusions, dementia, and paralysis o Late 1800s­proved to occur only in people who had contracted syphilis (STI) o Involves bacterial microorganisms invading brain tissue o Mental illness clearly due to physical cause  Treatment: medical/physical o Rest, good nutrition, exercise, stress reduction, and hospitalization o Electroconvulsive therapy (ECT) 1920s, reduces psychosis and depression o Medication (1950s) some sedatives reduce psychotic symptoms o Less useful treatments­ bloodletting, vomiting, lobotomies Early Psychological Tradition  Plato­ (ancient Greek) poor behavior is caused by bad experiences leading to faulty learning  Philippe Pinel/moral therapy­ (18 century France) advocated placing patients in an environmental that encouraged healthy social interactions and intellectual curiosity th  Dorothea Dix­ (19 century U.S.) increase public awareness of mental illness and value of hospitalization  Mental hygiene movement­ Mid­10 century effort to improve care of the mentally disordered by informing the public of their mistreatment Psychoanalytic Tradition  Structures of the mind o Id­ animal mind, responsible for primary processes, or basic drives (eating, sleeping, sex). Wants to maximize pleasure (pleasure principle). Unconscious o Superego ­ conscience, operates on moral principles o Ego ­ realistic mind, basis of secondary processes or logical thinking. Runs interference between the superego and the id Exam 1 Review Sheet 2  Defense mechanisms o Ego uses defense mechanisms to cope with anxiety created by conflict between the id and superego­  Displacement­ transferring an emotion from an unacceptable object to an acceptable one  Sublimation­ transforming unacceptable urges into acceptable interests/activities  Reactional formation ­ replacing an urge with its opposite  Denial­ refusing to recognize reality  Transference o Clients projects own thoughts/feelings/desires onto therapist  Countertransference o Therapist projects own thoughts/feelings/desires onto client  Psychosexual stages (Freud) o Oral stage (0­2)  Sexual gratification focused on the mouth  Fixation causes dependency or rebelliousness o Anal stage (2­3)  Sexual gratification focused on the anal sphincter  Fixation can lead to being too messy/careless or too neat/perfectionistic o Phallic stage (3­6)  Interested in own genitalia; conflicts with parents over masturbation  Oedipus/Electra complex­ become sexually attracted to the opposite­sex parent, wanting to eliminate same­sex parents as a rival  Fixated will have unhealthy relationships o Latency stage (school age)  Sexual interest goes underground until puberty o Genital stage (puberty)  Become involved in sexual relationships with appropriate others (final stage) o Explains adult personality as resulting from the success (or lack thereof) a child experiences navigating stages of sexual development. o Fixation results when development does not progress past a certain stage due to over or under gratification or urges  Oedipus/Electra complex o Sexually attracted to the opposite­sex parent, wanting to eliminate same­sex parents as a rival  Castration anxiety o Fear in young boys that they will be mutilated genitally because of their lust for their mothers  Spontaneous remission o Recovery that would have taken place without treatment Behavioral model  Classical conditioning­ (Pavlov, 1849­1936)­ when two things are experienced together, people form an association between them o Not necessarily conscious, automatic response (not a choice) o Explanation for phobias/anxiety Prepared learning­ humans biologically predisposed to learn some connections faster than others  o Uses in therapy  Classic extinctio­ experiences stimulus without feared event many times decreases fear  Exposure therapies­ experiencing the thing that makes you anxious while using relaxation skills will, over time, reduce anxiety  Aversive counterconditioning­ pair an unpleasant stimulus with an unwanted behavior to reduce the behaviors Exam 1 Review Sheet 3  Operant conditioning (B.F. Skinner)­ occurs when people learn a connection between a behavior and its consequences which leads them to make choices about behavior o Pleasant consequences­ behavior will be repeated o Unpleasant consequences­ behavior will be reduced o Used to explain behavior problems, addictions  Behavioral modification (contingency management) o Systematic use of rewards and/or punishments to shape behavior  Operant extinction o If reward is removed, behavior should diminish, possible after a e burst (brief increase in frequency)  Experimental neurosis o Behavior disorder produced experimentally o Produced in the laboratory by putting subjects in a situation where they are required to make discriminations or produce problems solving responses that are beyond capacity.  Seligman and learned helplessness o Animal that get shocked, but able to escape, will escape; animal unable to escape shocks won't try even when escape becomes possible Theory: feeling out of control can stop someone from trying to improve his/her situation (depression). o o Similar concept­ animals unable to read cues, and thus unable to predict what will happen Humanistic Model  Self­actualization­ people strive to achieve their highest potential against difficult life experiences  Person­centered therapy (Carl Rodgers) o Emphasizes importance of nconditional positive regard(therapist accepts client's feelings and actions without criticism, makes him/her feel understood and valued)  Motivational interviewing (Miller and Rollnick) o Increase client desire to change through use of empathic and supportive communication techniques Cognitive Model  Personal construct theory (George Kelly) o Employ methods of categorizing ourselves, others, and events to help us predict/understand the world o Can be overdone/misapplies (i.e. prejudice, depression) o Unusual experiences can use misunderstanding of the odds (i.e. trauma, substance abuse)  Cognitive Therapy (Aaron Beck) o Cognitive therapy­ change thoughts and emotions/behaviors will follow o Learning patterns of negative thinking and restructuring negative thoughts should reduce depression  Cognitive dissonance (Festiger) o When attitudes and cognitions are not consistent with behavior, we feel internal distress as cognitive dissonance o Treatment of behavior disorders may involve trying to induce cognitive dissonance  Systematic desensitization­ diminish excessive fears, involving gradual exposure to the feared stimulus paired with a positive coping experience, usually relaxation Explanations and treatment of abnormal behavior—  Biopsychosocial model o Approach to psychopathology (study of psychological disorders) o Mental disorders are due to the interactions between biological, psychological, and social forces  Bio­genes, chemistry, maturation.  Psycho­behavior, cognitions  Social­social network, environment, culture  Genes and psychological dysfunction Exam 1 Review Sheet 4 o Gene­ molecules of DNA found on chromosomes within cell nuclei  Affect a variety of characteristics (IQ, eye color, vulnerability to mental illness)  Major methods of research include twin and adoption studies o Most traits are lygenetic­ determined by a number of genes and outside influences  Developing a psychological disorder may be slightly affected by interactions among genes  Greater genetic link for some disorders (i.e. alcoholism, schizophrenia)  Diathesis­stress model o An individual inherits the genetic potential to develop a disorder, which emerges if certain conditions are met o Diathesis­ inherited vulnerability  Reciprocal gene­environment model o Just as the environment affects the expression of genes, genetic make­up affects a person's experience of his/her environment  Genotype vs. phenotype o Genotype­ set of traits contained in your genetic code o Phenotype­ traits actually expressed  Importance of twin studies o Comparison of twins with unrelated or less closely related individuals. If twins share common characteristics (disorder) even if they were reared in different environments, there strong evidence of genetic involvement  Central nervous system (CNS) o Brain and spinal cord  Peripheral nervous system (PNS) o Communicated between CNS and body o Somatic nervous system­ controls voluntary activity o Autonomic nervous system­ controls involuntary activity (heart, lungs, digestion). 2 parts work together to maintain homeostasis:  Sympathetic­ perceived threat triggers  Fight or flight (FFR): stimulates endocrine system, increases heart rate/respiration, degreases digestion/immune function  Parasympathetic­ restores body to state of rest/recuperation after arousal  Rest and digest  Structures of the brain o Extrapyramidal system­ neural network that is part of the motor system causing involuntary movements  Cerebellum­ coordinates movement  Reticular activating system (RAS)­ affect level of arousal, functioning impaired in ADHD, schizophrenia  Caudate nucleus­ directs motor behavior, impairment associated with OCD o Hypothalamus­ intrinsic behavior (sleep, hunger, sex, basic emotions). Runs the:  Endocrine system­ releases hormones into bloodstream ass part of FFR  Norepinephrine (NE)­ too little  depression, too much  schizophrenia/mania  Cortisol­ promotes healing short term, cause damage with long­term exposure o Limbic system­ found between brain stem and cerebral cortex  Amygdala­ emotionally­motivated behavior (aggression)  Hippocampus­ develops throughout life  Involved in memory (from perception to long term memory)  Chronic alcohol use causes damage o Frontal cortex­ executive functions  Complex functions like planning, reasoning, language, art, math  Involved in impulse control, attention self­monitoring of behavior, organization of information Exam 1 Review Sheet 5  Implicated in ADHD, OCD, schizophrenia, depression o Hypothalamic­pituitary­adrenocortical (HPA) axis  Set of direct influences and feedback interactions among three endocrine glands (in the name)  Controls reactions to stress and regulates many body processes (digestion, immune system, mood, emotions, sexuality)  Neurotransmitters (NT)­ chemicals that cause communications between neurons o Acetylcholine (Ach)­  Involved in movement (including in PNS), sleep, learning, and memory  Affected by many psychiatric medications, causing side effects such as dry mouth, constipation, retention of urine, nasal congestion, GI problems o Serotonin­  Affects information processing ­ regulated mood, behavior, and thought processes  High levels­ pro­social behavior and reduced sexual desire  Low levels­ depression and impulsive behavior (i.e. aggression, suicide, eating and sexual binges) o Dopamine­  Involved in movement/activation of other NT  High levels­ schizophrenia  Increases in response to rewards, both natural and chemical (substances of abuse) o GABA­  Inhibitory (low levels­ anxiety)  Maturation and psychopathology o Physical development­ due to maturation, with age disorders may…  Emerge­(schizophrenia) or remit­ (some ADHD)  Differ in expression­ types of fear are typical of certain ages  Emerge due to physical deterioration (i.e. dementia)  Systems theory­ o Viewing something as comprise of a number of interacting systems  Holism vs. reductionism o Seeing something as more than the sum of its parts; includes also interactions between part  Observational learning/social learning (Albert Bandura) o People learn from doing and observing others o Explains that exposure to TV violence increases aggression in children  Implicit memory o Condition of memory in which a person cannot recall past events despite acting in response to them Research methods, assessment, and diagnosis —know advantages and disadvantages of various types of research designs and methods of assessment. Foundations of the scientific method  Empirical – knowledge is gathered through observation  Objective – the researcher’s own biases should not influence the outcome of the investigation  Skeptical – have to be willing to examine other possible explanations, and to be willing to discard explanations that don’t fit  Replicable – later, similar studies find similar results  Steps in the process: o Formulate a question o Form a hypothesis – prediction about the result of an experiment o Test the hypothesis o Analyze data o Synthesize with other research o Develop new theories Exam 1 Review Sheet 6 Internal vs. external validity  Validity – extent to which a research technique accurately measures what it is intended to measure o Internal validity – extent to which the results of a study can be attributed to the intended manipulation of a variable (related to amount of control) o External validity – extent to which the results of a study can be generalized (applied) to the larger population. Enhanced by… - Collecting a large sample – increases chance sample group will be similar to population - Collecting a randomized sample – selecting subjects from a population in such a way that each subject has an equal chance of ending up in the sample Generalization  Extent to which research can be applied to the larger, general population Randomized sample  Collecting a randomized sample – each subject has an equal chance of ending up in the sample Confounding factors  Confounding factors – any factor other than the intended manipulation that causes or masks the effects of a study o Threatens the validity and/or reliability of research results o Just being in treatment can be a confound (gives hope/support) Descriptive vs. experimental designs  Descriptive design – examination of a phenomenon or group of phenomena; no manipulation of variables  Experimental design – involves manipulation of one or more variables to investigate causation Correlational designs—limits  Correlation does not prove causation Case studies  Case study – detailed analysis focused on one event, individual, or group. Can be either descriptive or experimental o Advantages - Communicates a lot of information - Because of high level of scrutiny can get good internal validity o Disadvantages - Can be misleading because not every case is representative of a problem, impairing generalizability Independent vs. dependent variable  Independent variable – factor manipulated by the experimenter  Dependent variable – factor measured to determine if it is affected by the change in the independent variable Experimental vs. control groups  Experimental group – subjects who receive the intervention  Control group­ subjects who do not receive the intervention o Compared to the experimental group to see if the intervention produced a difference Randomization of subjects  Assigning subjects to research conditions so that each subject has an equal chance of ending up in either the experimental or the control group o Advantages – enhances internal validity, reduces the risk that other factors are creating any possible effects o Disadvantages - Randomization and control groups can be hard to get - Can be impractical/unethical to withhold treatment - If a control group is not an option, may use a quasi­experimental design Quasi­experimental designs  Subjects are not randomly assigned to conditions but a manipulation is made Hypothesis  Prediction about the result of an experiment Analogue model Exam 1 Review Sheet 7  Approach to research that employs subjects who are similar to clinical clients, allowing replication of a clinical problem under controlled conditions Ethical principles in research  Ethical Principles in Research o DO NO HARM o Informed consent – individuals must be aware of: - What will take place during the study - Any risks involved - Freedom to withdraw at any time Meta­analysis  Involves combining results from many studies to explore whether a specific treatment is associated with a higher rate of change than spontaneous remission or placebo effect Characteristics of evidence­supported treatments  Evidence­supported treatments: o Specific interventions determined by meta­analyses to have a high chance of success o With some exception, most effective psychotherapies are cognitive­behavioral interventions Placebo effect  Placebo effect – recovery due to the mere expectation that healing will occur Importance of therapeutic relationship  Most important factor in treatment success is quality of therapeutic alliance – working relationship between therapist and client Criticisms of diagnosis  Criticisms of diagnosis: o Labeling is not fixing – knowing you have bipolar disorder does not cure your bipolar disorder o Association with the medical model – suggests the client is diseased o Lack of reliability – different clinicians working with the same case may arrive at different diagnoses o Diagnoses may be stigmatizing – may feed existing prejudices towards individuals Diagnostic and statistical manual  Diagnostic and Statistical Manual (DSM 5)  Published in May of 2013 with: o Goal of enhancing clinical utility of diagnostic system more so than other concerns such as research/legal use o Reliance on research base to improve validity o Focus on dimensional as well as categorical approach to diagnosis, recognizing that many psychological problems present along a continuum of severity Reliability  Reliability – extent to which measurement is consistent (e.g., across time and across evaluators) Approaches to classification—categorical, dimensional, and prototypical Familial aggregation  Extent to which a disorder would be found among patient’s relatives Comorbidity  Presence of two or more disorders in an individual at the same time Clinical interview/mental status exam  Clinical interview – general assessment of current and past functioning. Includes: o Personal history – family, social, educational, medical, legal o Current functioning o Psychological symptoms o May include a mental status exam – assesses observed aspects of current functioning, such as: - Orientation – to time, place, person, and situation - Emotional state Exam 1 Review Sheet 8 - Quality of thought processes and perception Standardization  Standardization – process involves: o Establishment of norms – determining how a representative sample of the population scores on the measure o Development of uniform rules of use for a measure to enable reliable: - Administration of the instrument - Evaluation of an individual’s responses - Comparisons to other individuals’ responses Behavioral assessment  Behavioral assessment – systematic measurement of client’s thoughts, feelings, and behaviors o Antecedents – what is going on before the behavior o Behaviors – the target behavior and any that accompany it o Consequences – what happens as a result of the behavior Self­monitoring  Self­monitoring – client performs own behavioral assessment Reactivity  Reactivity – process by which a behavior changes because it is being measured o Can reduce the validity of findings (e.g., client behaves better because he’s being monitored) o Clinically can be useful because often moves a behavior in the desired direction by itself Projective testing  Subject projects unconscious conflicts onto ambiguous stimuli o Emerged from the psychodynamic approach o Thematic Apperception Test (TAT) – subject asked to make up a story about some of a series of 31 cards depicting ambiguous scenes o Rorschach  Not standardized, thus: o Reliability is low – e.g., different examiners give a test different ways, producing different results o Validity is also minimal – e.g., hard to know what we’re seeing if can’t compare it to others MMPI—basic ideas behind standard scales and validity scales  Minnesota Multiphasic Personality Inventory (MMPI): o Standard scales – measure aspects of personality and psychopathology (e.g., depression, antisocial behavior) o Validity scales – measure/compensate for response patterns that would otherwise distort findings (e.g., minimizing, exaggerating) Intelligence quotient (IQ)  Score on an intelligence test estimating a person’s deviation from average test performance Neuropsychological testing  Measure brain function through performance on behavioral or cognitive tasks o Neuropsychological Assessment Battery – 33 tests assesses cognitive functioning in adults when brain injury is suspected - Nicely reliable and valid - Less affected by language differences - Can take a long time to administer Neuroimaging techniques—CT, MRI, PET  Allows non­intrusive examination of nervous system structure and function  CT­ Computerized axial tomography, locate brain tumors, injuries, and other structural and anatomical abnormalities o Involve repeated radiation  MRI­ magnetic resonance imaging, doesn’t use X­rays, head placed in magnetic field and radio waves are transmitted o Claustrophobic for patients Exam 1 Review Sheet 9  PET­ positron emission tomography, measure functioning brain, hot spots picked up by isotopes in the brain and blood, learn which portions of the brain are working or not EEG  Electroencephalogram, measure electrical activity patterns in the brain, taken through electrodes placed on the scalp Psychophysiological assessment  Psychophysiological assessment – measure physical changes thought to reflect emotional processes o Galvanic Skin Response – intended to measure anxiety through sweat - Less opportunity for bias/faking - Concerns with validity:  Not everything leaves a physical trace  Subject to reactivity Rorschach  Rorschach Inkblot Test: o Subject asked what he/she sees in a series of 10 inkblots o Information about the subject gathered from overall themes (e.g., lots of blood or mothers) Psychotic disorders— for each disorder know characteristics, general facts about prevalence (e.g., whether it’s common or not), course, causes, and treatment. Positive symptoms  Positive symptoms – presence of something unusual/unexpected o Hallucinations – most common are auditory hallucinations o Delusions – strongly­held beliefs that persist despite the presence of conflicting evidence Negative symptoms—avolition, alogia, anhedonia, asociability, diminished emotional expression  Negative symptoms – absence of something usual/expected o Avolition – significant difficulty initiating and persisting in goal­directed activity o Alogia – poverty of speech o Anhedonia – lack of enjoyment o Asociability – avoiding company of others o Diminished emotional expression Types of delusions  Delusions of reference – belief that the statements or actions of others refer to you  Grandiose delusions – exaggerations of one’s importance  Somatic delusions – involving health Catatonia  Catatonia – grossly disorganized or abnormal motor behavior possibly involving: Disorganized symptoms  Disorganized symptoms – disruptions in normal functioning o Disorganized speech – jumps from topic to topic or is completely garbled o Inappropriate affect – expressed emotion that doesn’t match the situation o Disorganized behavior – inappropriate to the setting o Catatonia – grossly disorganized or abnormal motor behavior possibly involving:  Physical rigidity  Excessive activity  Meaningless repetition of others’ words or movements Schizophreniform disorder  Schizophreniform disorder – symptoms of schizophrenia with duration less than six months but more than one Brief psychotic disorder  Brief psychotic disorder – symptoms of schizophrenia lasting at least a day but less than a month Exam 1 Review Sheet 10 Delusional disorder  Delusional disorder – presence of delusions but usually not other psychotic symptoms Physical causes of schizophrenia and other types of psychosis  Hypothesized to be primarily physical – tend to have decreased brain volume, blood flow, limbic system activity, and brain symmetry  Genetic link – 48% risk will develop if identical twin has it  Prenatal/early childhood insults – e.g., maternal malnutrition or viral illness causes damages that emerges later  Development gone awry – normal process of pairing down unnecessary synapses gets out of hand  Dopamine hypothesis – medications that reduce dopamine also reduce psychotic symptoms  Causes of psychosis (other than substances) o Substances of abuse – e.g., cocaine and amphetamines o Medical problems – e.g., brain tumors, MS o Severe psychological disorders – e.g., severe depression (including postpartum), even eating disorders (delusions about body) Effects of stress on psychotic symptoms  Stressful life events can lead people with a predisposition for schizophrenia to develop the full blown disorder  Stress is linked to schizophrenia (living in a big city, tragic event) Expressed emotion  Hostility, criticism, and over­involvement demonstrated by some families toward a family member with a psychological disorder, often can contribute to relapse Concept of functional recovery  Goal of treatment often is functional recovery – developing skills to reduce impact of disorder, not curing the disorder Prodromal stage  Period of 1 or 2 years before serious symptoms of schizophrenia occur but when less severe yet unusual behaviors start to appear Dopamine receptor antagonists  Dopamine receptor antagonists – typical antipsychotics o Reduce positive symptoms, not negative o Long­term use can cause extrapyramidal side effects, including tardive dyskinesia Tardive dyskinesia  Tardive dyskinesia: strange, involuntary movements of tongue, jaw, trunk, or limbs Serotonin­dopamine antagonists  Serotonin­dopamine antagonists (atypical antipsychotics) o Fewer extrapyramidal side effects o Can cause metabolic syndrome Metabolic syndrome  Involving risk of hyperglycemia, diabetes, and increased cholesterol Types of psychosocial treatments  Psychosocial – generally are adjunctive o Social learning/token economies – desired behaviors (e.g., medication compliance) rewarded with credits exchanged for privileges o Cognitive­behavioral therapy – reduces positive symptoms and re­hospitalization rates o Social skills training – commonly focused on basic skills o Family therapy – reduces relapse rates o Assertive community treatment – involves ongoing case management integrated with psychiatric care o Supported employment o Cognitive remediation – focused on improving learning and memory through repeated practice of basic skill Deinstitutionalization movement Exam 1 Review Sheet 11  Deinstitutionalization: o Movement begun in 1960s to transfer chronically mentally ill patients to community placements o Estimated 1/3 to 2/3 of homeless have schizophrenia o Only about half of individuals with schizophrenia are currently receiving treatment

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ISBN: 9780321947345

Since the solution to 78 from 8.5 chapter was answered, more than 232 students have viewed the full step-by-step answer. The answer to “Series of squares Prove that if gak is a convergent series of positive terms, then the series gak 2 also converges.” is broken down into a number of easy to follow steps, and 21 words. The full step-by-step solution to problem: 78 from chapter: 8.5 was answered by , our top Calculus solution expert on 12/23/17, 04:24PM. This full solution covers the following key subjects: . This expansive textbook survival guide covers 128 chapters, and 9720 solutions. This textbook survival guide was created for the textbook: Calculus: Early Transcendentals, edition: 2. Calculus: Early Transcendentals was written by and is associated to the ISBN: 9780321947345.

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Series of squares Prove that if gak is a convergent series