How to solve the geometric series convergence problem 59

Key takeaway: When a series looks messy, the first thing to check is whether every term is just a power of the same expression. Here, it is. That makes the rest straightforward.

For problem 59, $\sum_{n=1}^{\infty}(-5)^n x^n,$ combine the powers: $(-5)^n x^n = (-5x)^n.$ So this is a geometric series with ratio $r=-5x.$

Convergence

A geometric series converges exactly when $|r|<1.$ Thus $|-5x|<1\quad\Rightarrow\quad |x|<\frac15.$ So the series converges for $-\frac15<x<\frac15.$

Sum

Because the series begins at $n=1$, $\sum_{n=1}^{\infty} r^n=\frac{r}{1-r}.$ With $r=-5x$, $\sum_{n=1}^{\infty}(-5x)^n=\frac{-5x}{1+5x}.$

Answer

• Convergence interval: $\left(-\frac15,\frac15\right)$
• Sum: $\frac{-5x}{1+5x}$

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Pitfalls the pros know 👇 Don’t test the endpoints separately and assume the series might converge there anyway. For a geometric series, once $|r|=1$, it does not converge. Also, make sure you keep the starting index in mind: starting at $n=1$ changes the sum formula.

What if the problem changes? If the same pattern appeared as $\sum_{n=0}^{\infty}(-5x)^n$, the sum would become $\frac{1}{1+5x}$ instead. If the exponent were on $x$ only, like $\sum (-5)^n x^{n+1}$, then you would factor out one extra $x$ before using the geometric-series formula.

Tags: geometric series, convergence interval, series sum formula