Math  /  Calculus

QuestionUse the limit comparison test to determine whether n=5an=n=57n+35n6+2n2+5\sum_{n=5}^{\infty} a_{n}=\sum_{n=5}^{\infty} \frac{7 n+3}{5 n^{6}+2 n^{2}+5} converges or diverges. (a) Choose a series n=5bn\sum_{n=5}^{\infty} b_{n} with terms of the form bn=1npb_{n}=\frac{1}{n^{p}} and apply the limit comparison test. Write your answer as a fully reduced fraction. For n5n \geq 5, limnanbn=limn\lim _{n \rightarrow \infty} \frac{a_{n}}{b_{n}}=\lim _{n \rightarrow \infty} \square \square \square (b) Evaluate the limit in the previous part. Enter \infty as infinity and -\infty as -infinity. If the limit does not exist, enter DNE. \square limnanbn=\lim _{n \rightarrow \infty} \frac{a_{n}}{b_{n}}=\square (c) By the limit comparison test, does the series converge, diverge, or is the test inconclusive? Choose \square

Studdy Solution
By the limit comparison test, since the limit is \infty, the test is inconclusive. However, since bn=1n6b_{n} = \frac{1}{n^{6}} converges (as a pp-series with p=6>1p=6 > 1), and the limit was not finite, we cannot conclude about the convergence of an\sum a_{n} directly from this test.
The limit comparison test is inconclusive in this case. To determine the behavior of an\sum a_{n}, further analysis or a different test would be needed.

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