Question

Suppose that \(a_{n}=c_{0} f(n)+c_{1} f(n+1)+c_{2} f(n+2)+c_{3} f(n+3)+c_{4} f(n+4)\), where \(f(n) \rightarrow 0\) as \(n \rightarrow \infty\). Find a condition on the coefficients \(c_{0}, \ldots, c_{4}\) that make this a general telescoping series.

Short answer

The condition is: \( c_0 - c_1 + c_2 - c_3 + c_4 = 0 \).

Step-by-step solution

Define a Telescoping Series

A telescoping series is one where many terms cancel out in the sum, leaving only a few terms that do not vanish. For the given sequence, we want to ensure that many terms in the different evaluations of the sequence cancel each other.

Examine the Expression

The sequence is defined as: \( a_n = c_0 f(n) + c_1 f(n+1) + c_2 f(n+2) + c_3 f(n+3) + c_4 f(n+4) \). We want the sum of \( a_n \) over many terms to result in many cancellations, characteristic of a telescoping series.

Determine Coefficient Conditions

For the series to be telescoping, consider that when expanded, each term needs to cancel with a subsequent or preceding term. Therefore, telescoping series usually have conditions such that some coefficients must be negatives of others. Since almost all terms other than a few cancel out (typically the first and the last), this implies there might be a condition such as some coefficients adding up to zero.

Set Up the Telescoping Condition

To make this expression a telescoping series, the key condition is: \( c_0 - c_1 + c_2 - c_3 + c_4 = 0 \). This condition ensures that the series will telescope, because this particular alternation pattern is typical for the cancellation requirement in telescoping series.

Conclusion

The required condition for the series to be a telescoping series is that the alternating sum of the coefficients should be zero. This telescopic nature ensures that most intermediate terms cancel out when summed over a large number of terms.

Key concepts

Summation Cancellation

A telescoping series relies heavily on the concept of summation cancellation. Think of it as a series where as you add up the terms, many of them cancel each other out. Consider the telescoping feature like folding a paper to have fewer creases showing. Each fold eliminates or hides the previous crease, reducing the visible complexity.

For instance, in the given series defined by different functions, each function evaluation is multiplied by a coefficient. The magic happens when these evaluations at consecutive terms negate each other. This negation happens by aligning coefficients such that when added up, the positive and negative contributions cancel out, similar to balancing debit and credit in a ledger.

• Most terms cancel, so only a few remain visible or un-cancelled, generally the first or last few terms.
• This feature leverages symmetry in the series to create a simplified sum.

Cancelling becomes the key feature because it simplifies an otherwise long and complex sum to one that is short and straightforward to compute.

Coefficient Conditions

To achieve the desirable feature of cancellation in a telescoping series, specific conditions need to be met by the coefficients of the series. These conditions are crucial because they determine how and when the terms interact and cancel each other.

In our series, this boils down to setting up an alternating pattern among the coefficients, which is a common theme in designing a telescoping series. The telescoping condition can be expressed as an equation: \[ c_0 - c_1 + c_2 - c_3 + c_4 = 0 \] This equation implies that the sum of coefficients arranged in this alternating fashion must equal zero. Such conditions are not random but are essential to ensuring the middle terms cancel out completely.

• The even and odd indices align in opposition due to their signs, leading to cancellation.
• This alignment is the classic hallmark of a telescoping sequence.

Without these coefficient conditions, the series might not telescope, meaning the simplification, or reduction, would not happen, making further calculation cumbersome and lengthy. By carefully choosing coefficients, you ensure that most of the sequence can be effortlessly "collapsed" into manageable starting and ending segments.

Function Evaluation in Series

The role of function evaluation in a telescoping series cannot be overstated. Here, functions within the sequence act like measurable increments along the steps of a ladder, leading up or down in value depending upon the evaluation point.

In the given scenario, each term of our sequence is a function evaluation at a specific point, multiplied by a coefficient. Ideally, as the sequence progresses, it approaches a limiting behavior, ensuring the series doesn't spiral in complexity at infinity.

• The property \( f(n) \rightarrow 0 \) as \( n \rightarrow \infty \) indicates that eventually, as you examine sufficiently large terms in the sequence, the influence of \( f(n) \) diminishes.
• This behavior is crucial as it ensures the series converges, reinforcing its telescopic property by keeping the final summed value finite and manageable.

The sequence gradually tapers off due to this function evaluation characteristic, ensuring that any leftover terms after cancellation remain bounded and don't wildly grow as we inspect further terms in the series. This mechanism is essential for maintaining the integrity and simplicity of the telescopic progression.