Question

(a) Show the following limit of a sequence: lim n?? (n + ln(n)) / n = 1. (b) Use the limit comparison test to show that ? n=1 ? (n + ln(n)) / n^3 converges. Hint: ((n + ln(n)) / n^3) / (1 / n^2) = (n + ln(n)) / n^3 * n^2 = (n + ln(n)) / n

          (a) Show the following limit of a sequence:
lim n?? (n + ln(n)) / n = 1.
(b) Use the limit comparison test to show
that ? n=1 ? (n + ln(n)) / n^3 converges.
Hint: ((n + ln(n)) / n^3) / (1 / n^2) = (n + ln(n)) / n^3 * n^2 = (n + ln(n)) / n

(a) Show the following limit of a sequence:
lim n?? (n + ln(n)) / n = 1.
(b) Use the limit comparison test to show
that ? n=1 ? (n + ln(n)) / n^3 converges.
Hint: ((n + ln(n)) / n^3) / (1 / n^2) = (n + ln(n)) / n^3 * n^2 = (n + ln(n)) / n

Added by Anthony B.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Adi S

11/22/2023

Step 1

We can rewrite the sequence as $\frac{n}{n}+\frac{\ln(n)}{n} = 1+\frac{\ln(n)}{n}$. Now, as $n$ approaches infinity, the term $\frac{\ln(n)}{n}$ approaches 0, since the natural logarithm function grows slower than the linear function. Show more…

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(a) Show the following limit of a sequence: lim n->∞ (n + ln(n)) / n = 1. (b) Use the limit comparison test to show that ∑ (n=1 to ∞) (n + ln(n)) / n^3 converges. Hint: ((n + ln(n)) / n^3) / (1 / n^2) = (n + ln(n)) / n^3 * n^2 = (n + ln(n)) / n