Show that the series converges. What is the value of the series? sum_{n=2}^infty(-frac{5}{3})^n(frac{2}{5})^{n+1}

ka1leE 2020-10-20 Answered
Show that the series converges. What is the value of the series?
n=2(53)n(25)n+1
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Expert Answer

ensojadasH
Answered 2020-10-21 Author has 100 answers

We have to Show that the series converges. What is the value of the series?
Series is given below:
n=2(53)n(25)n+1
We will use geometric series test .
Geometriic series test says if series of the form arn1 and |r|<1 then we can say that series is converges and sum of series is given by S.
Work is shown below:
n=1arn1
|r|<1
S=a1r
Now compare the given series to geometric series and find value of r .
If r<1 then series converges.
In this case r=23<1 so series converge.
After that find value of a (starting term of series).
With the help of a, r and sum of geometric series formula we will find sum.
Work is shown below:
n=2(53)n(25)n+1
n=2(53)n(25)n+1
n=225(1)n(23)n
|r|=|23|
|r|=23<1
Previous step continue
n=225(1)n(23)n
84516135+32405...
a=845
r=23
S=8451+23
=875
Answer
S=875

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Jeffrey Jordon
Answered 2021-12-16 Author has 2064 answers

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