Question

How do I find the limit of a series?

Answer 1

Hint: In this problem, we have to explain how to find the limit of a series. We can first know what the limit of a series is. The limit of a series is the value, the series terms are approaching as \[n \to \infty \]. Here we do not have any rule to find the limit of the series. But there are specific types of series for which we can compute the limit.

Complete step by step solution:
We can now explain what the limit of a series is.
 The limit of a series is the value, the series terms are approaching as \[n \to \infty \].
We can now take an example.
We can now find the limit of the series \[\sum\limits_{n=1}^{\infty }{\dfrac{{{2}^{3n}}}{{{64}^{\dfrac{n}{2}}}}}\].
To find the limit of the series, we will identify the series as \[{{a}_{n}}\], and then take the limit of \[{{a}_{n}}\] as \[n \to \infty \].
\[\begin{align}
  & \Rightarrow \displaystyle \lim_{n \to \infty }{{a}_{n}}=\displaystyle \lim_{n \to \infty }\dfrac{{{2}^{3n}}}{{{64}^{\dfrac{n}{2}}}} \\
 & \Rightarrow \displaystyle \lim_{n \to \infty }\dfrac{{{8}^{n}}}{{{\left( \sqrt{64} \right)}^{n}}} \\
 & \Rightarrow \displaystyle \lim_{n \to \infty }\dfrac{{{8}^{n}}}{{{8}^{n}}} \\
 & \Rightarrow \displaystyle \lim_{n \to \infty }1 \\
 & \Rightarrow 1 \\
\end{align}\]
Therefore, the limit of the series is 1.
We should know that, whenever a limit exists in a series, then the series is convergent or summable. In this case the limit is called the sum of the series.. Otherwise the series is said to be divergent.

Note: Students should remember that the notation of limit should be with n tends to infinity. We should also remember that we do not have any rule to find the limit of the series. But there are specific types of series for which we can compute the limit. We should also know that, limit of series and sum of series are different.