Explain why or why not Determine whether the

Chapter 1, Problem 1RE

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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

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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

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