Question

Find all values of x for which the series converges. For these values of x, write the sum of the series as a function of x. sum_{n=1}^infty(3x)^n

Series
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asked 2021-02-25
Find all values of x for which the series converges. For these values of x, write the sum of the series as a function of x.
\(\sum_{n=1}^\infty(3x)^n\)

Answers (1)

2021-02-26

Consider the series
\(\sum_{n=1}^\infty(3x)^n\)
Consider a G.P series \(\sum_{n=1}^\infty r^n\)
Series is convergent for \(|r|<1\)
Sum of this infinite series is given by
\(S_\infty=\frac{a}{1-r}\)
\(\sum_{n=1}^\infty(3x)^n=(3x)^1+(3x)^2+(3x)^3+(3x)^4....\)
\(\sum_{n=1}^\infty(3x)^n=(3x)+(3x)^2+(3x)^3+(3x)^4....\)
The given series is a Geometric progression with the
first term \(=3x\)
common ratio \(=3x\)
The given series is convergent for
\(|3x|<1\)
\(\Rightarrow-1<3x<1\)
\(\Rightarrow-\frac13\)
Sum of the series \(\sum_{n=1}^\infty(3x)^n\) for \(-\frac13\)
Here
\(a=3x,r=3x\)
\(S_\infty=\frac{a}{1-r}\)
\(S_\infty=\frac{3x}{1-3x}\)
Answer:
Series is convergent for \(-\frac13\)
Sum of the series is \(\frac{3x}{1-3x}\)

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