Question

Find the sum of the convergent series. sum_{n=0}^infty5(frac23)^n

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asked 2021-02-15
Find the sum of the convergent series.
\(\sum_{n=0}^\infty5(\frac23)^n\)

Answers (1)

2021-02-16
To find the sum of the convergent series: \(\sum_{n=0}^\infty5(\frac23)^n\)
Solution:
Expanding the given series, we get
\(\sum_{n=0}^\infty5(\frac23)^n=5(\frac23)^1+5(\frac23)^2+5(\frac23)^3+...\)
\(=5[(\frac23)^1+(\frac23)^2+(\frac23)^3+...]\)
Now, taking the series \((\frac23)^1+(\frac23)^2+(\frac23)^3+...\)
Here,we can find that sequence is in geometric progression.
First term is \(a_1=\frac23\)
Common ratio is:
\(r=\frac{(\frac23)^2}{(\frac23)}\)
\(=\frac23\)
Sum of infinite terms of G.P. is given as:
\(S=\frac{a_1}{1-r}\)
Sum of the sequence \((\frac23)^1+(\frac23)^2+(\frac23)^3+...\) will be:
\(S=\frac{\frac23}{1-\frac23}\)
\(=\frac{\frac23}{\frac13}\)
\(=2\)
Now, sum of the series \(\sum_{n=0}^\infty5(\frac23)^n\) will be:
\(\sum_{n=0}^\infty5(\frac23)^n=5[(\frac23)^1+(\frac23)^2+(\frac23)^3+...]\)
\(=5\cdot2\)
\(=10\)
Hence, required sum is 10.
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