Does the series (showing the picture) converge or diverge? Choose the correct answer below. 1) The integral test shows that the series converges 2) The nth-term test shows that the series converges 3) The series diverges because the series is a geometric series with |r|>=1 4) The nth-term test shows that the series diverges sum_{n=1}^inftyfrac{4^n}{n+1}

Question
Series
asked 2021-01-16
Does the series (showing the picture) converge or diverge?
Choose the correct answer below.
1) The integral test shows that the series converges
2) The nth-term test shows that the series converges
3) The series diverges because the series is a geometric series with |r|>=1
4) The nth-term test shows that the series diverges
\(\sum_{n=1}^\infty\frac{4^n}{n+1}\)

Answers (1)

2021-01-17
The given series is \(\sum_{n=1}^\infty\frac{4^n}{n+1}\)
Determine whether the series \(\sum_{n=1}^\infty\frac{4^n}{n+1}\) converges or diverges as follows.
The nth term test:
If the limit \(\lim_{n\rightarrow\infty} a_n\) either does not exist or not equal to zero, then \(\sum_{n=1}^\infty a_n\) diverges.
Find the limit \(\lim_{n\rightarrow\infty} a_n\) as shown below.
\(\lim_{n\rightarrow\infty} a_n=\lim_{n\rightarrow\infty}\frac{4^n}{n+1}\)
\(\lim_{n\rightarrow\infty} a_n=\lim_{n\rightarrow\infty}\frac{4^n}{n(1+\frac1n)}\)
\(\lim_{n\rightarrow\infty} a_n=\lim_{n\rightarrow\infty}\frac{\frac{4^n}{n}}{n(1+\frac1n)}\)
\(\lim_{n\rightarrow\infty} a_n=\frac{\lim_{n\rightarrow\infty}\frac{4^n}{n}}{\lim_{n\rightarrow\infty}n(1+\frac1n)}\)
\(\frac{\infty}{1}\)
\(=\infty\)
Since the limit \(\lim_{n\rightarrow\infty}\frac{4^n}{n+1}\) does not exist, the series \(\sum_{n=1}^{\infty}\frac{4^n}{n+1}\) diverges.
Hence, the nth term test shows that the series diverges.
Thus, the correct option is 4.
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