Question

By shifting the index of summation as in equation \((9)\) or \((13)\), rewrite the given power series so that the general term involves \(t^{n}\). $$ \sum_{n=2}^{\infty} n(n-1) a_{n} t^{n-2} $$

Short answer

Question: Rewrite the power series \(\sum_{n=2}^{\infty} n(n-1) a_{n} t^{n-2}\) such that the general term involves \(t^{n}\) instead of \(t^{n-2}\). Answer: The rewritten power series is \(\sum_{m=0}^{\infty} (m+1)(m+2) a_{m+2} t^{m}\).

Step-by-step solution

Shifting the index

We will shift the index by introducing a new index, \(m = n-2\), so that the general term involves \(t^{m}\) instead of \(t^{n-2}\). We need to express \(n\) in terms of \(m\), which is simply \(n = m + 2\).

Updating the summation limits

Now, we need to update the summation limits based on this shift. For the lower summation limit, when \(n=2\), now we have \(m = 2 - 2 = 0\). For the upper summation limit, since it is \(\infty\), it remains the same.

Rewriting the general term

With the shifted index and new summation limits, we can rewrite the general term of the power series. Replace \(n\) with \(m+2\) and \(t^{n-2}\) with \(t^{m}\) in the general term: $$ (n-1)(n)a_n t^{n-2} = (m+1)(m+2)a_{m+2} t^{m} $$

Rewriting the power series

Finally, we can rewrite the entire power series with the updated general term and the new summation limits: $$ \sum_{n=2}^{\infty} n(n-1) a_{n} t^{n-2} = \sum_{m=0}^{\infty} (m+1)(m+2) a_{m+2} t^{m} $$ Now, the given power series is rewritten with the general term involving \(t^{m}\) instead of \(t^{n-2}\).

Key concepts

Index of Summation

In the world of mathematics, the index of summation is the variable that represents the position of a term within a series. When we deal with power series, this index is critical because it gets tied to the powers of the variable we are summing over.

Think of the index as the 'running counter' in a summation. Usually denoted by letters like n, i, or m, it changes value with each term of the sum. For instance, in the sum \(\sum_{n=1}^{5} a_{n}\), n is the index of summation that takes on the values 1 through 5 to sum the respective terms a₁, a₂, ..., until a₅.

When we shift the index, like we did in the original exercise, we're simply changing the 'name' and the starting point of this counter. By setting m equal to n-2, we've created a new starting point and a new way to reference each term in the series, facilitating the manipulation of the series to fit a desired form, such as having a power of t^m instead of t^n-2.

Summation Limits

The summation limits determine where the summation starts and ends. The lower limit indicates the initial value of the index of summation, while the upper limit shows where the summation stops, which might be a finite number or even infinity.

In our example, the original series has lower and upper summation limits of 2 and \(\infty\), respectively, indicated by \(\sum_{n=2}^{\infty}\). After shifting the index, these limits need to be updated to \(m = 0\) and \(\infty\), expressed as \(\sum_{m=0}^{\infty}\), which reflects the change in the index of summation. When the upper limit is infinity, it signifies an infinite series, one that has infinitely many terms.

This update in the summation limits after the index shift is crucial for preserving the integrity of the series. It ensures that we sum over the correct terms and that our series still represents the function or sequence we are interested in analyzing or manipulating.

General Term of Power Series

A power series is a series where each term involves a variable (often x or t) raised to a power. The general term of a power series gives us a formula that describes the nth term in the series.

The general term is often represented as \(a_{n}x^{n}\), where \(a_{n}\) is a coefficient that might depend on n, and \(x^{n}\) signifies that the variable x is being raised to the power equal to the current index of summation. In the given exercise, the power series had a more complex general term \(n(n-1) a_{n} t^{n-2}\).

By shifting the index and updating the summation limits, we can rewrite the general term to involve \(t^{m}\) instead. The process requires substituting n with m+2 to match our new index and adjust the powers of t accordingly. This manipulation is not just symbolic; it can simplify calculations and help in studying the convergence properties of the power series, which are pivotal in the realms of calculus and complex analysis.