We propose to infinite series: 2 - 4/3 + 8/9 - 16/27 + Write the previous series

Question

We propose to infinite series: 2 - 4/3 + 8/9 - 16/27 + Write the previous series in a summation notation Prove the convergence of this series and compute its value Check the convergence of the series: Let S = Sigma a_n be a convergent series and S_N Partial sum. Prove that a_n is a convergent sequence and provide its limit If a_n > 0 discuss the monotonicity of S_N and provide a lower and upper bounds Consider the following series: Sigma^infinity (-1)^n n/n^2 + 1 Prove the convergence of the series. Provide an approximation with an error less than 0.01 of the series value.

Explanation / Answer

1. 2 - 4/ 3 + 8/ 9 -16/27 + ....

Here common ratio = -4/3/2=-2/3

Therefore the previous term is 2/-2/3 =-3

Here the common ratio is less than 1 , so it converges .

S= a/(1-r) = 2/(1 + 2/3) = 6/5

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