# 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
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
$$\sum_{n=1}^\infty\frac{4^n}{n+1}$$

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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