How to find the sum of a telescoping series?
Understand the Problem
The question is asking for a method or procedure to calculate the sum of a telescoping series, which is a specific type of series where many terms cancel each other out. The approach would typically involve identifying the pattern of the series and using it to derive a formula for the sum.
Answer
The sum of a telescoping series is given by $S = f(1) - f(N+1)$.
Answer for screen readers
The sum of the telescoping series can be calculated as:
$$ S = f(1) - f(N+1) $$
Steps to Solve
- Identify the telescoping series pattern
Examine the given series to find a general term, usually represented as $a_n$. A telescoping series can often be expressed as a difference of two fractions that will cancel out as the series progresses.
- Write the series explicitly
Express the first few terms of the series to visualize the cancellations. For example, if the series is given in the form:
$$ S = \sum_{n=1}^{N} (f(n) - f(n+1)) $$
Write out the terms such as $f(1) - f(2), f(2) - f(3), \ldots, f(N) - f(N+1)$.
- Observe the cancellations
When you write out the series, notice that many terms will cancel each other. For example:
$$ S = f(1) - f(N+1) $$
Only the first and last terms will remain after cancellation.
- Calculate the sum
Now plug in the limits of the series into your expression to find the total sum. For the telescoping series, the sum can be simplified to:
$$ S = f(1) - f(N+1) $$
Where $f(1)$ and $f(N+1)$ are now evaluated based on the original function.
- Conclude the solution
Make sure to state the final result as the sum of the series you were initially trying to compute.
The sum of the telescoping series can be calculated as:
$$ S = f(1) - f(N+1) $$
More Information
Telescoping series are particularly useful in summation problems because they greatly reduce complexity by canceling out many terms. This simplifies the evaluation, making it easier to compute finite sums.
Tips
- Failing to correctly identify the terms that cancel out can lead to an incorrect sum.
- Not evaluating the first and last terms correctly, which can produce errors in the final answer.
