Question

A general telescoping series is one in which all but the first few terms cancel out after summing a given number of successive terms. Let \(a_{n}=f(n)-2 f(n+1)+f(n+2)\), in which \(f(n) \rightarrow 0\) as \(n \rightarrow \infty .\) Find \(\sum_{n=1}^{\infty} a_{n}\).

Short answer

The sum is \(f(1)\).

Step-by-step solution

Identify the Series Structure

The problem asks to find the sum of the series \(\sum_{n=1}^{\infty} a_n\) where \(a_n=f(n)-2f(n+1)+f(n+2)\). The given function \(f(n)\) fulfills the condition that \(f(n) \rightarrow 0\) as \(n \rightarrow \infty\).

Analyze the Telescoping Nature

Notice that \(a_n = f(n) - 2f(n+1) + f(n+2)\) can be rewritten to emphasize the cancellation property: \(a_n = [f(n) - f(n+1)] - [f(n+1) - f(n+2)]\). This structure suggests that when summed, each \(f(n+1)\) cancels with \(-f(n+1)\) of the subsequent term.

Write Out the First Few Terms

Let's write out the first few terms to observe the cancellation:- When \(n=1\), \(a_1 = f(1) - 2f(2) + f(3)\).- When \(n=2\), \(a_2 = f(2) - 2f(3) + f(4)\).- When \(n=3\), \(a_3 = f(3) - 2f(4) + f(5)\).Continuing this way, we see that each term \(f(n+1)\) within one \(a_n\) is cancelled by the term \(-f(n+1)\) in \(a_{n+1}\).

Sum and Simplify

Using the telescoping property, most terms cancel out after summing up the series:\[\sum_{n=1}^{N} a_n = (f(1) - f(2)) + (f(2) - f(3)) + \,\text{...}\, + (f(N+1) - f(N+2))\]Upon summing all the terms, we notice all intermediate terms cancel and we are left with: \[\sum_{n=1}^{N} a_n = f(1) - f(N+2).\]

Evaluate the Limit

To find \(\sum_{n=1}^{\infty} a_n\), consider the limit as \(N \to \infty\).Since \(f(n) \to 0\) as \(n \to \infty\), we conclude \[\lim_{N \to \infty} f(N+2) = 0.\]Therefore, the sum becomes:\[\sum_{n=1}^{\infty} a_n = f(1) - 0 = f(1).\]

Conclusion

The series converges to \(f(1)\) because all intermediate terms cancel out in a telescoping manner. Only \(f(1)\) remains, with \(f(n) \to 0\) eliminating the effect of any remaining terms.

Key concepts

Series Convergence

Understanding series convergence is an essential aspect of evaluating infinite series. Convergence refers to whether a series sums to a finite value as we add more and more terms. It's like asking, "Does the series settle on a number, or does it keep growing forever?" This is crucial in mathematics because only convergent series provide meaningful, finite results.
For many series, especially telescoping series, certain properties help determine convergence easily. In our specific telescoping series \(\sum_{n=1}^{\infty} a_n\), we rely on the fact that the terms will eventually cancel out as we sum them up. Moreover, the behavior of the function, such as \(f(n) \to 0\) as \(n \to \infty\), ensures that the series doesn't blow up to infinity. Instead, it converges to a specific value, which we found to be \(f(1)\).

To verify convergence:

• Check for a pattern where terms cancel out, simplifying the series.
• Ensure any remaining terms tend to zero as \(n\) becomes very large.

Once these checks are satisfied, as with the telescoping series, you can confidently claim that the series converges.

Function Behavior at Infinity

The behavior of functions as they approach infinity is often highlighted as one of the critical aspects when dealing with series. Specifically, understanding \(f(n) \to 0\) as \(n \to \infty\) is important for analyzing series. As we deal with infinite sums, knowing how individual terms behave as \(n\) increases is crucial for predicting the eventual outcome of the series.

For the given series, \(f(n)\) shrinking to zero ensures that any contributions from terms such as \(f(N+2)\) become negligible. This is why, when summing the series, we ultimately only need to consider the initial term \(f(1)\). As \(N\) grows eternally, the terms at the tail end of our series approach zero, leading them to have minimal impact on the sum. This property is an assurance of the series' convergence to a value determined mainly by the behavior of the initial terms.
Understanding a function's behavior at infinity allows us to:

• Predict whether an infinite sum will produce a finite result.
• Recognize when terms in a series will become insignificant as we continue to add more.

These insights are invaluable for effective series evaluation.

Telescoping Property

The telescoping property is a fascinating concept, particularly useful in simplifying complex series. This property refers to the idea that many terms in the series cancel out successively, leaving only a few to be considered for finding the sum. It's somewhat like having a giant accordion where pulling it causes all sections to collapse into a single portion.

In our original problem, the term \(a_n = f(n) - 2f(n+1) + f(n+2)\) highlights the telescoping nature. By rearranging, we recognize pairs like \([f(n) - f(n+1)] - [f(n+1) - f(n+2)]\), where most parts cancel out after summation. Writing the first few terms reveals this cancellation:

• \(a_1 = f(1) - 2f(2) + f(3)\)
• \(a_2 = f(2) - 2f(3) + f(4)\)
• \(a_3 = f(3) - 2f(4) + f(5)\)

As seen, successive terms lead to cancellations, leaving us eventually with just \(f(1) - f(N+2)\).
The telescoping property allows us to:

• Drastically simplify series by focusing only on surviving terms.
• Identify patterns where terms effectively "disappear" from the series sum.

Leveraging this property effectively aids in determining the sum of complex infinite series swiftly.