Question

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

Short answer

Question: Verify the following identity: $$\sum_{k=0}^{\infty} a_{k+1} x^{k}+\sum_{k=0}^{\infty} a_{k} x^{k+1}=a_{1}+\sum_{k=1}^{\infty}\left(a_{k+1}+a_{k-1}\right) x^{k}$$ Answer: After combining the given series, rewriting the sum in terms of a general series, making use of the given equation, and comparing the expressions, we have verified the given identity to be true: $$\sum_{k=0}^{\infty} a_{k+1} x^{k}+\sum_{k=0}^{\infty} a_{k} x^{k+1}=a_{1}+\sum_{k=1}^{\infty}\left(a_{k+1}+a_{k-1}\right) x^{k}$$

Step-by-step solution

Combine the given series

First, we need to combine both series into a single expression. We can do this by noticing that both series have an x^k term, so we can factor x^k out: $$\sum_{k=0}^{\infty}\left(a_{k+1} x^k + a_{k} x^{k+1}\right)$$

Rewrite the sum in terms of a general series

Next, rewrite the sum as the sum of k terms of the series: $$a_1x^0 + \sum_{k=1}^{\infty}\left(a_{k+1} x^k + a_{k}x^{k+1}\right)$$

Make use of the given equation

Now use the given equation and replace the series terms with the given equation: $$a_1x^0 + \sum_{k=1}^{\infty} \left(a_{k+1}+a_{k-1}\right) x^{k}$$

Compare the two expressions

Compare the expression we obtained in Step 3 with the given equation and see if they are equal: $$a_1x^0 + \sum_{k=1}^{\infty} \left(a_{k+1}+a_{k-1}\right) x^{k} = a_{1}+\sum_{k=1}^{\infty}\left(a_{k+1}+a_{k-1}\right) x^{k}$$

Verify the identity

Both sides of the equation have the same structure. The coefficient of x^0 is the same, and the coefficients of x^k for k ≥ 1 are also the same. Therefore, the given equation is true: $$\sum_{k=0}^{\infty} a_{k+1} x^{k}+\sum_{k=0}^{\infty} a_{k} x^{k+1}=a_{1}+\sum_{k=1}^{\infty}\left(a_{k+1}+a_{k-1}\right) x^{k}$$

Key concepts

Power Series Methods

A power series is a way to represent a function as an infinite sum of terms. It's like expanding a function into a series of simpler components. In the context of differential equations, power series methods help solve equations that can't easily be solved using standard techniques. When we talk about using power series to solve differential equations, we are referring to expressing solutions as infinite sums that approximate functions over certain intervals. This approach is particularly useful for equations where solutions can't be neatly expressed as polynomials or basic functions. When you work with power series, each term in the series is typically expressed as \( a_{k}x^{k} \), where \(k\) is an index running from zero to infinity, and \(a_{k}\) are coefficients that depend on the function being approximated or solved for. The series converges when, for certain values of \(x\), adding more terms results in values that get closer to a specific limit. Determining these coefficients and ensuring the series converges properly are key parts of using power series methods effectively.

Infinite Series

Infinite series are sums that have an endless number of terms. These are not just theoretical constructs; they play an essential role in mathematics, especially in calculus and analysis. The infinite series are often used to express complicated functions in a more manageable form, especially when dealing with differential equations. By breaking down these equations into simpler series, mathematicians can more easily understand and solve them. There are specific tests, like the ratio test or root test, to determine if an infinite series converges or diverges. Convergence means that the series approaches a specific value as more and more terms are added. Divergence, on the other hand, implies that the series does not settle to a finite limit. In differential equations, ensuring that the series converges is vital for accurately describing a function or a solution over a desired interval.

Verification of Series Solutions

Verification of series solutions involves checking that our power series representation accurately solves the original equation. This involves comparing the series form with the equation to ensure they match. In practice, verifying a series solution means substituting the series into the equation, performing necessary algebraic manipulations, and demonstrating both sides of the equation are equal. In our given exercise, this process involves combining like terms and ensuring the series equate. Key points in the verification process include:

• Ensuring coefficients match for each power of \(x\).
• Checking the validity of algebraic manipulations throughout the solution.
• Comparing every term in the series with the corresponding term in the original equation.

Verifying the equation is crucial as it confirms that the series representation correctly addresses the problem posed by the differential equation. It's easy to make mistakes with indices or coefficients, so careful step-by-step analysis is essential in the verification process.