Questions: Determine if the series below is a telescoping series.
[
sumn=1^inftyleft(10^n+2+19^n+3right)
]
Select the correct answer below:
The series is telescoping.
The series is not telescoping.
Transcript text: Determine if the series below is a telescoping series.
\[
\sum_{n=1}^{\infty}\left(10^{n+2}+19^{n+3}\right)
\]
Select the correct answer below:
The series is telescoping.
The series is not telescoping.
Solution
Solution Steps
Step 1: Identify the Series
We are given the series
\[
\sum_{n=1}^{\infty}\left(10^{n+2}+19^{n+3}\right).
\]
Step 2: Analyze the Terms
The terms of the series can be expressed as \(10^{n+2}\) and \(19^{n+3}\). We need to check if these terms can cancel each other out when the series is expanded.
Step 3: Determine Cancellation
In a telescoping series, most terms cancel out with subsequent terms. However, in this case, the terms \(10^{n+2}\) and \(19^{n+3}\) do not have any terms that will cancel with them in the series.
Step 4: Conclusion
Since there are no cancellations among the terms, we conclude that the series is not telescoping.