Verify that the infinite series converges. ∑n=0∞(−0.2)n=1−0.2+0.04−0.008+…

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Verify that the infinite series converges.  ∑n=0∞(−0.2)n=1−0.2+0.04−0.008+…

Nicole Conner · Accepted Answer

We have to verify that series converges:  ∑n=0∞(−0.2)n=1−0.2+0.04−0.008+…  Looking at series we can observe that  −0.21=−0.2  −0.04−0.2=−0.2  −0.008−0.04=−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∞=a1−r  Where, a is first term and r is common ratio of series.  Given series:  1−0.2+0.04−0.008+…  Therefore,  first term, a=1  and common difference or common ratio, r=−0.2  Now applying sum formula for infinite geometric series,  s∞=a1−r  =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.

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

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