Question
Decide whether you think the series converges or diverges, and if ti converges, estimate the sum.$$\frac{1}{5}+\frac{2}{5^{2}}+\frac{3}{5^{3}}+\frac{4}{5^{4}}+\cdots$$
Step 1
The given series is $$\frac{1}{5}+\frac{2}{5^{2}}+\frac{3}{5^{3}}+\frac{4}{5^{4}}+\cdots$$ This can be written in the form of a sum as: $$\sum_{n=1}^{\infty} \frac{n}{5^n}$$ This is an infinite series where each term is of the form $\frac{n}{5^n}$. Show more…
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