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.

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.