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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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