Example of Series such that $\sum a_n$ converges but $\sum a_n^4$ diverges

Question

I am interested in finding

Example of Series such that $\sum a_n$ converges but $\sum a_n^4$ diverges

I am able to find converse like if $\sum a_n^4$ converges but $\sum a_n$ diverges using harmonic series

But I not able to find suitable example for above

Any help will be appreciated

Answer 1

Hint: Try something alternating, e.g. $$ a_n = \frac{(-1)^n}{\sqrt[4]{n}}. $$

Answer 2

When $|a_n|<1$ you have that $|a_n|^4 < |a_n|$. Therefore, the convergence of $\sum |a_n|$ guarantees the convergence of $\sum |a_n|^4$.

Hence, to answer your question you must use negative terms. One approach is to use alternating series.

For example, consider the series of the sequence

$$a_n = (-1)^n \frac{1}{\sqrt[4]{n}} $$

such that $a_n^4 = 1/n$ gives rise to the Harmonic series.

The convergence of $\sum a_n$ can be established (without finding the exact sum) using the alternating series test, whereas $\sum a_n^4$ is the Harmonic series which diverges.