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


For example,


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.

&\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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