# Series Converges to b: Find Method

• Swerting
In summary, the series in question converges using the Ratio Test and can be found using the Taylor Polynomial for sin(x).
Swerting

## Homework Statement

I have the series $$\sum\frac{b^{(2n+2)}(-1)^{n}}{(2n+2)!}$$ from n=0 to infinity. I am trying to find what it converges to in terms of b.

## Homework Equations

Using the Ratio Test I have established that it does converge.

## The Attempt at a Solution

I have scoured the internet, my notes, and all my books, but I can't seem to find a way to find what these kinds of series (power I believe) converge to, only ways to see if they converge or not. I just need to find out the method to calculate what it converges to in terms of b. Thank you for any assistance.

$$\sum\frac{b^{(2n+2)}(-1)^{n}}{(2n+2)!}=\sum\frac{b^{(2n+2)}(-1)^{n}}{(2n+2)(2n+1)!}$$, from which you can differentiate term by term

Why would I differentiate it? How does that help find what it converges to?

$$\frac{d}{dx}\sum\frac{b^{(2n+2)}(-1)^{n}}{(2n+2)(2n+1)!}=\sum\frac{b^{(2n+1)}(-1)^{n}}{(2n+1)!}$$ which looks a bit like which function?

It looks like the general term of the Taylor Polynomail for sin(x)...so it is sin(b)?

The derivative of the series converges to sin(b), not the original series.

## 1. What does it mean for a series to converge?

Convergence refers to the behavior of a series as the number of terms increases towards infinity. A series converges if the sum of its terms approaches a finite value as the number of terms increases.

## 2. How can I determine if a series converges to a specific value?

The most common method is to use a convergence test, which is a mathematical tool that evaluates the behavior of a series and determines if it converges to a specific value. Examples of convergence tests include the ratio test, the root test, and the integral test.

## 3. Can a series converge to more than one value?

No, a series can only converge to one specific value. If a series has multiple limit points, it is considered divergent.

## 4. Is there a general method for finding the value a series converges to?

No, there is no one-size-fits-all method for finding the value a series converges to. The convergence test used will depend on the specific series being evaluated.

## 5. Why is it important to determine if a series converges?

Determining if a series converges is important in many branches of science and mathematics, as it can help verify the accuracy of calculations and make predictions about the behavior of systems. Convergent series also have useful applications in areas such as engineering and physics.

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