Verify that the infinite series converges. sum_{n=0}^infty(-0.2)^n=1-0.2+0.04-0.008+...

banganX

banganX

Answered question

2021-03-06

Verify that the infinite series converges.
n=0(0.2)n=10.2+0.040.008+

Answer & Explanation

Nicole Conner

Nicole Conner

Skilled2021-03-07Added 97 answers

We have to verify that series converges:
n=0(0.2)n=10.2+0.040.008+
Looking at series we can observe that
0.21=0.2
0.040.2=0.2
0.0080.04=0.2
Ratio of last terms and previous terms are same hence it is geometric infinite series.
We know that if we have geometric infinite series a+ar+ar2+ar3+ then its sum formula is given by
s=a1r
Where, a is first term and r is common ratio of series.
Given series:
10.2+0.040.008+
Therefore,
first term, a=1
and common difference or common ratio, r=0.2
Now applying sum formula for infinite geometric series,
s=a1r
=11(0.2)
=11+0.2
=11.2
=1012
=0.83
We know that if sum of series is less than one then series is known as convergence.
Here sum of series is 0.83 which is less than one.
Hence we can say that given series are converges.

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?