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Convergence condition

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    Vu Hung
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Problem Statement

Consider the geometric series

∑n=0∞7(m−13)n.\sum_{n=0}^{\infty} 7\left(\frac{m-1}{3}\right)^n.

Find all real values of mm for which the series converges, and then state its sum.

Hints

Use the condition ∣r∣<1|r|<1 for convergence of an infinite GP.

Solutions

Here

r=m−13.r=\frac{m-1}{3}.

Convergence requires

∣m−13∣<1  ⟹  ∣m−1∣<3  ⟹  −2<m<4.\left|\frac{m-1}{3}\right|<1 \implies |m-1|<3 \implies -2<m<4.

For these values,

S∞=71−m−13=214−m.S_\infty=\frac{7}{1-\frac{m-1}{3}}=\frac{21}{4-m}.

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