 # 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 opatovaL 2021-01-16 Answered

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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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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