# Does the series (showing the picture) converge or diverge? Choose the correct answer below. 1) The integral test shows that the series converges 2) Th

Does the series (showing the picture) converge or diverge?
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
$\sum _{n=1}^{\mathrm{\infty }}\frac{{4}^{n}}{n+1}$

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doplovif
The given series is $\sum _{n=1}^{\mathrm{\infty }}\frac{{4}^{n}}{n+1}$
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}\frac{{4}^{n}}{n+1}$ converges or diverges as follows.
The nth term test:
If the limit $\underset{n\to \mathrm{\infty }}{lim}{a}_{n}$ either does not exist or not equal to zero, then $\sum _{n=1}^{\mathrm{\infty }}{a}_{n}$ diverges.
Find the limit $\underset{n\to \mathrm{\infty }}{lim}{a}_{n}$ as shown below.
$\underset{n\to \mathrm{\infty }}{lim}{a}_{n}=\underset{n\to \mathrm{\infty }}{lim}\frac{{4}^{n}}{n+1}$
$\underset{n\to \mathrm{\infty }}{lim}{a}_{n}=\underset{n\to \mathrm{\infty }}{lim}\frac{{4}^{n}}{n\left(1+\frac{1}{n}\right)}$
$\underset{n\to \mathrm{\infty }}{lim}{a}_{n}=\underset{n\to \mathrm{\infty }}{lim}\frac{\frac{{4}^{n}}{n}}{n\left(1+\frac{1}{n}\right)}$
$\underset{n\to \mathrm{\infty }}{lim}{a}_{n}=\frac{\underset{n\to \mathrm{\infty }}{lim}\frac{{4}^{n}}{n}}{\underset{n\to \mathrm{\infty }}{lim}n\left(1+\frac{1}{n}\right)}$
$\frac{\mathrm{\infty }}{1}$
$=\mathrm{\infty }$
Since the limit $\underset{n\to \mathrm{\infty }}{lim}\frac{{4}^{n}}{n+1}$ does not exist, the series $\sum _{n=1}^{\mathrm{\infty }}\frac{{4}^{n}}{n+1}$ diverges.
Hence, the nth term test shows that the series diverges.
Thus, the correct option is 4.
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