The terms of a series are defined recursively. Determine the convergence or divergence of the series. Explain your reasoning. sum_{n=1}^infty a_n a_1=frac15, a_{n+1}=frac{cos n+1}{n}a_n

Jason Farmer 2021-02-05 Answered
The terms of a series are defined recursively. Determine the convergence or divergence of the series. Explain your reasoning.
n=1an
a1=15, an+1=cosn+1nan
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Mayme
Answered 2021-02-06 Author has 103 answers

Consider the given terms for the infinite series. Find the ratio of the (n+1)th and nth term. Use the ratio test to evaluate the nature of the series.
a1=15
Now,  an+1=cosn+1nan
an+1an=cosn+1n
Let L=limnan+1an
Further, simplify using the ratio test. Hence, from the ratio test, the series converges.
So, L=limncosn+1n
=limncosn+1nnn
=limncosnn+1n1
=limn(cosnn+1n)
=limn(cosnn)+limn(1n)
0+1 [From squeeze theorem limn(cosnn)=0]
0+0
So, L=0
Thus, L<1
Hence, n=1an converges.

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Jeffrey Jordon
Answered 2021-12-17 Author has 2064 answers

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