Solution Found!
Explain why or why not Determine whether the
Chapter 1, Problem 1RE(choose chapter or problem)
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. The terms of the sequence \(\left\{a_{n}\right\}\) increase in magnitude, so the limit of the sequence does not exist.
b. The terms of the series \(\sum 1 / \sqrt{k}\) approach zero, so the series converges.
c. The terms of the sequence of partial sums of the series \(\sum a_{k}\) approach 5/2, so the infinite series converges to 5/2.
d. An alternating series that converges absolutely must converge conditionally
Questions & Answers
QUESTION:
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. The terms of the sequence \(\left\{a_{n}\right\}\) increase in magnitude, so the limit of the sequence does not exist.
b. The terms of the series \(\sum 1 / \sqrt{k}\) approach zero, so the series converges.
c. The terms of the sequence of partial sums of the series \(\sum a_{k}\) approach 5/2, so the infinite series converges to 5/2.
d. An alternating series that converges absolutely must converge conditionally
ANSWER:STEP_BY_STEP SOLUTIONStep-1 Definition ; A series is said to be convergent if it approaches some limit .Formally , the infinite series a n is convergent if the sequence of partial sums n=1 n S n a k is convergent . Conversely , a series is divergent if the k=1sequence of partial sums is divergent. If U and V are convergent series , then k k( U +kV k ) and ( U - Vk k ) are convergent . If C = / 0 , then U k and V kboth converge or both diverge . Step-2 a) Now , we have to check the result ‘The terms of the sequence {a } incneases in magnitude , so the limit of the sequence does not exist ‘ is true (or) false. For example ; a = 1 - 1 . n n 1 Clearly as n , then the value of n 0 So , we can say that {a } increases in magnitude , and lim ( 1 - 1 ) n n n = lim (1) - li