Math Problem Statement

Find the sum of the series: ∑ (−1)^n 4^(n+2) / (2n+1)! from n=0 to infinity.

Solution

The series given in the image is:

n=0(1)n4n+2(2n+1)!\sum_{n=0}^{\infty} \frac{(-1)^n 4^{n+2}}{(2n+1)!}

To solve this, we can start by simplifying the general term and recognizing its structure. First, let’s factor out the 424^2 term to simplify:

42=16,so we rewrite the series as16n=0(1)n4n(2n+1)!4^2 = 16, \quad \text{so we rewrite the series as} \quad 16 \sum_{n=0}^{\infty} \frac{(-1)^n 4^n}{(2n+1)!}

This series looks similar to the Maclaurin series expansion for sin(x)\sin(x), which is:

sin(x)=n=0(1)nx2n+1(2n+1)!\sin(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}

In our case, we have powers of 44 instead of xx. Comparing the two, we see that the sum can be written as:

sin(2)=n=0(1)n22n+1(2n+1)!\sin(2) = \sum_{n=0}^{\infty} \frac{(-1)^n 2^{2n+1}}{(2n+1)!}

To match the structure, replace 22 with 44:

16sin(4)16 \cdot \sin(4)

Thus, the sum of the given series is:

16sin(4)16 \cdot \sin(4)

You can compute the final value of 16sin(4)16 \sin(4) using a calculator to get a numerical result.

Let me know if you'd like further details or a step-by-step breakdown!

Here are 5 related questions to explore:

  1. How does the Maclaurin series expansion for cos(x)\cos(x) differ from sin(x)\sin(x)?
  2. Can you generalize this series for higher powers of 4?
  3. What other common functions have similar Taylor or Maclaurin expansions?
  4. How does the factorial term in the denominator affect convergence?
  5. What is the radius of convergence for a typical sine series?

Tip: Recognizing patterns in series, such as the sine or cosine expansions, is a useful skill when working with infinite sums!

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Infinite Series
Maclaurin Series
Sine Function

Formulas

Maclaurin series expansion for sin(x): ∑ (-1)^n x^(2n+1) / (2n+1)!
General form of sine series: sin(x) = ∑ (-1)^n x^(2n+1) / (2n+1)!

Theorems

Maclaurin series expansion

Suitable Grade Level

Undergraduate Mathematics