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Evaluating series Evaluate the following

Chapter 1, Problem 14RE

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QUESTION:

12-20. Evaluating series Evaluate the following infinite series or state that the series diverges.

\(\sum_{k=0}^{\infty}\left(-\frac{1}{5}\right)^{k}\)

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QUESTION:

12-20. Evaluating series Evaluate the following infinite series or state that the series diverges.

\(\sum_{k=0}^{\infty}\left(-\frac{1}{5}\right)^{k}\)

ANSWER:

STEP_BY_STEP SOLUTION Step-1 Definition ; A series is said to be convergent if it approaches some limit .Formally , the infinite series a n is convergent if the sequence of partial sums n=1 n S n a k is convergent . Conversely , a series is divergent if the k=1sequence ofpartial sums is divergent.NOTE : The terms grow without bound , so the sequence does not converge. If U and V are convergent series , then ( U + V ) and ( U

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