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(- n ) n (-

(- n ) n (-

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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can you please answer both questions. Decide whether those two series will converge. Thank you

(- n ) n (-
Transcribed Image Text:(- n ) n (-
Expert Solution
Step 1

Given series

(-1)" logn n 1 To prove: If the given series converges
Step 2

Note: We use the Ratio test which is given as

an+1 L lim n co an If L<1, then the series converges If L 1, then the series diverges. If L 1 or the limit fails to exist, then the test is inconclusive.
Step 3

Applying the rule,

(-1)n+1 log(n1) n+1 (-1)n log n an+1 lim n-o an lim n co (-1)"+1 log(n1) lim (-1) log n n1 noo (-1)" (1) log(n + 1) lim n n 1 (-1) log(n 1) n 1 (-1)" log n nco n lim logn n 0o log(n1) - lim n 1 log n n-co log(n1) =-lim n co n 1 logn log(n 1) lim =-lim n 1 n-00 logn noo
Step 4

We solve it individually.

On applying the limit, we get the indeterminant form, thus we use the L'Hospital Rule i.e. differentiate the numerator and differentiate the denominator and then take the limit. 1 1 n1 = -lim *lim 1/n 1 n o0 n-oo = 0 Hence, the series converges
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