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explain why the ratio test cannot be used on the series1+15+110+150+1100+1500+....then show that the series converges and find its sum.

Short Answer

Expert verified

The required sum to converges the series is 43

Step by step solution

01

Given information

Consider the given series,

1+15+110+150+1100+1500+.....

To explain why the ratio test cannot be used on the given series. to show that the series converges.

02

Calculation

Consider the series and rewrite it,

According to the ratio test โˆ‘k=1โˆžakbe the series with all terms positive and L=limkโ†’โˆž(ak+1ak)then,

1. if L<1 series converges

2. if L>1 series diverges

if L= 1 the test is inconclusive

Since L=limkโ†’โˆžak+1akdoes not exists this implies that the ratio test cannot be used.

03

Further Calculation.

According to the convergence and divergence of geometric series.

1. for r<1, the geometric series โˆ‘k=0โˆžrkconverges to the sum 11-r

2. for rโ‰ฅ1, the geometric series โˆ‘k=0โˆžrkdiverges.

04

Simplification

Now the series,1+110+1100+....+15+150+1500+...โ†’(1).

Rewrite the given series as follows,

1+110+1100+....=โˆ‘k=0โˆž110k151+110+1100+....=15โˆ‘k=0โˆž110k

Hence the series (1) can be written as,

1+110+1100+....+15+150+1500+...=โˆ‘k=0โˆž110k+15โˆ‘k=0โˆž110k

Hence the series is convergent as its sum of two convergent geometric series.

05

Find the sum of the obtained series.

The sum of the convergent geometric series is 11-r.

hence, the seriesโˆ‘k=0โˆž110kwillconvergeto11-110.

Hence the sum of the series can be calculated as follows,

โˆ‘k=0โˆž110k+15โˆ‘k=0โˆž110k=109+15109=1091+15=10965=43

Hence the sum of the series is 43.

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