Question

Verify the given equation. $$ \sum_{n=0}^{\infty} a_{n}(x-1)^{n+1}=\sum_{n=1}^{\infty} a_{n-1}(x-1)^{n} $$

Short answer

Question: Verify that the following equation is true: $$ \sum_{n=0}^{\infty} a_{n}(x-1)^{n+1}=\sum_{n=1}^{\infty} a_{n-1}(x-1)^{n} $$ Answer: We can verify the given equation by changing the indices of the left side series to start at \(n=1\), simplifying the series terms, and showing that the terms of both series are equal. Therefore, we can conclude that the equation is true: $$ \sum_{m=0}^{\infty} a_{m+1}(x-1)^{m+2} = \sum_{n=1}^{\infty} a_{n-1}(x-1)^{n} $$

Step-by-step solution

Analyze the given equation

First, let us analyze the equation. $$ \sum_{n=0}^{\infty} a_{n}(x-1)^{n+1}=\sum_{n=1}^{\infty} a_{n-1}(x-1)^{n} $$ We see that on the left side, the series starts at \(n = 0\) and on the right side, the series starts at \(n = 1\). Our goal is to change the left side series to start at \(n=1\) and adjust the terms accordingly.

Change the indices of the series

To change the left side series summation to start from \(n=1\), we need to replace variable \(n\) with a new variable, for example \(m = n-1\). Then, when \(n=0 \Rightarrow m=-1\). Therefore, when \(n=1\), so \(m=0\). Moreover, when \(m = n-1\), then when \(m+1=n\). So we have: $$ \sum_{m=1-1}^{\infty} a_{m+1}(x-1)^{(m+1)+1}=\sum_{n=1}^{\infty} a_{n-1}(x-1)^{n} $$ Now, our left side starts at \(n=1\), so we can rewrite our equation as: $$ \sum_{m=0}^{\infty} a_{m+1}(x-1)^{(m+1)+1}=\sum_{n=1}^{\infty} a_{n-1}(x-1)^{n} $$

Simplify the series terms

Now, let's simplify the left side series terms. $$ \sum_{m=0}^{\infty} a_{m+1}(x-1)^{(m+1)+1} = \sum_{m=0}^{\infty} a_{m+1}(x-1)^{m+2} $$

Compare the series

Now that we have both series in the same form, we can compare their terms. Notice that for every natural number \(m\) in the left side series, \(a_{m+1}(x-1)^{m+2}\), the terms are equal to the terms at the right side series, \(a_{n-1}(x-1)^{n}\) , for every natural number \(n\). We've verified our given equation: $$ \sum_{m=0}^{\infty} a_{m+1}(x-1)^{m+2} = \sum_{n=1}^{\infty} a_{n-1}(x-1)^{n} $$

Key concepts

Series Index Manipulation

One of the core techniques in dealing with mathematical series is manipulating the index of summation. This often involves changing the starting index of a series to facilitate comparison or simplification. Here, the goal was to make both series start at the same index.

To do this, we introduce a substitution where we let a new variable, say \(m\), take over the role of the old variable \(n\). For example, by setting \(m = n - 1\), we adjust the indices to align both series properly.

• Original index transformation: From \(n = 0\) to \(m = -1\) and from \(n = 1\) to \(m = 0\).
• This method ensures that when the series start at \(n = 1\), the terms are now aligned.

Ultimately, such manipulation helps in ensuring that both sides of an equation represent the same mathematical object, allowing for easier comparison and verification.

Infinite Series Verification

Verifying that two infinite series are equal, especially after index manipulation, involves checking that each corresponding term in the two series is identical. With the indices aligned, we can compare the series directly.

If a transformation is done correctly, the terms \(a_{m+1}(x-1)^{m+2}\) on the left should exactly match \(a_{n-1}(x-1)^{n}\) on the right.

• Check whether each function of the original variable matches after adjustment.
• The powers of expressions \((x-1)\) should also correspond directly for each term of the series.

By ensuring these matches, we confirm that both series represent the same sum—thus verifying that they are equivalent. This is a crucial step in proofs involving infinite series.

Power Series Expansion

Power series are a robust tool for representing a function as an infinite sum of terms involving powers of a variable, often denoted as \((x-a)\). In this problem, we deal with powers of \((x-1)\).

The term \((x-1)^{n}\) appears in the series, where \(n\) changes as per the index manipulation. Understanding power series involves recognizing how each term contributes to the function's behavior around a specified point, known as the expansion point.

• Each term \((x-1)^n\) demonstrates the individual power presentation of series.
• Adjusting indices mainly influences coefficients \(a_n\), which weigh the contribution of each power term.

By expanding a function into a power series, we can accurately approximate behaviors near given points, making calculations more manageable in analytical settings.