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The limit of a series is the value a series approaches as the number of terms approaches infinity. If the limit is infinite, the series is divergent. If it is finite, it is called convergent.

Calculating the limit of a seres

If the series is geometric, or in the form

$ \sum_{k=0}^\infty ar^k $

it will be convergent if $ -1<r<1 $ . The sum will be equal to

$ \frac{a}{1-r} $

For example,

$ \sum_{k=0}^\infty\dfrac{1}{2^k}=1+\frac12+\frac14+\frac18+\cdots=2 $

If the series is not geometric, it may not be possible to calculate it directly. One way to estimate it is to break the sum into two parts.

$ \sum_{n=1}^\infty a_n=\sum_{n=1}^k a_n+\sum_{n=k+1}^\infty a_n $

The first part of the sum can be calculated directly, while the second part can be estimated by using improper integrals.

$ \begin{align}&f(n)=a_n\\ &\int\limits_{k+1}^\infty f(x)dx\le\sum_{n=k+1}^\infty a_n\le\int\limits_k^\infty f(x)dx\end{align} $

The larger $ k $ is, the more accurate the estimation will be.

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