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?
\(\sum_{n=2}^\infty(-\frac{5}{3})^n(\frac{2}{5})^{n+1}\)

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Expert Answer

ensojadasH
Answered 2020-10-21 Author has 16248 answers

We have to Show that the series converges. What is the value of the series?
Series is given below:
\(\Rightarrow\sum_{n=2}^\infty(-\frac{5}{3})^n(\frac{2}{5})^{n+1}\)
We will use geometric series test .
Geometriic series test says if series of the form \(ar^{n-1}\) and \(|r|<1\) then we can say that series is converges and sum of series is given by S.
Work is shown below:
\(\Rightarrow\sum_{n=1}^\infty ar^{n-1}\)
\(|r|<1\)
\(\Rightarrow S_{\infty}=\frac{a}{1-r}\)
Now compare the given series to geometric series and find value of r .
If \(r<1\) then series converges.
In this case \(r=\frac23 <1\) 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:
\(\Rightarrow\sum_{n=2}^\infty(-\frac53)^n(\frac25)^{n+1}\)
\(\Rightarrow\sum_{n=2}^\infty(-\frac53)^n(\frac25)^{n+1}\)
\(\Rightarrow\sum_{n=2}^\infty\frac25(-1)^n(\frac23)^n\)
\(\Rightarrow|r|=|-\frac23|\)
\(|r|=\frac23<1\)
Previous step continue
\(\Rightarrow_{n=2}^\infty\frac25(-1)^n(\frac23)^n\)
\(\Rightarrow\frac{8}{45}-\frac{16}{135}+\frac{32}{405}...\)
\(\Rightarrow a=\frac{8}{45}\)
\(\Rightarrow r=-\frac{2}{3}\)
\(\Rightarrow S_{\infty}=\frac{\frac{8}{45}}{1+\frac23}\)
\(=\frac{8}{75}\)
Answer
\(\Rightarrow S_{\infty}=\frac{8}{75}\)

Not exactly what you’re looking for?
Ask My Question
46
 
content_user
Answered 2021-12-16 Author has 11052 answers

Answer is given below (on video)

0

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more
...