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# Prove some properties of the Bessel functions of the first kind of orders zero and one

We define the Bessel functions of the first kind of orders zero and one by 1. Prove that both and converge for all .
2. Prove that .
3. If we define two new functions prove that .

1. Proof. For the order zero Bessel function of the first kind we have and so using the ratio test we have Hence, converges for all .

For we have and so Hence, converges for all 2. Proof. We compute the derivative of directly, 3. Proof. First, we have On the other hand we have, Therefore, Hence, we indeed have # Compute the sum of the series ∑ (x-1)n / (n+2)!

Find all such that the series converges and compute the sum.

First, we apply the ratio test, for all . Therefore, the series converges for all . Next, we compute the sum for , In the case then all of the terms except the term are 0, and that term is ; hence, the sum is .

# Compute the sum of the series ∑ xn / (n+3)!

Find all such that the series converges and compute the sum.

First, we apply the ratio test for all . Therefore this converges for all . Furthermore, we compute the sum, for (since we’ll want to divide by in here), If then the sum is 0.

# Compute the sum of the series ∑ ((-1)n x2n) / n!

Find all such that the series converges and compute the sum.

By the ratio test we have for all . Thus, the series converges for all . Furthermore, we compute the sum from the series expansion of , # Compute the sum of the series ∑ (((-1)n) / (2n+1)) * (x/2)2n

Find all such that the series converges and compute the sum.

First, we have Then, applying the ratio test we have Thus, the series converges for all with which implies . Furthermore, if , then the series converges as the alternating harmonic series. Then, we compute the sum for , # Compute the sum of the series ∑ (2n xn) / n

Find all such that the series converges and compute the sum.

First, we apply the ratio test, Thus, the series converges for . For the boundary, if then this is the alternating harmonic series which converges. If then it is the harmonic series, which diverges. Therefore, the series converges for all . Then, we compute, # Compute the sum of the series ∑ (-2)n ((n+2) / (n+1)) xn

Find all such that the series converges and compute the sum.

First, we apply the ratio test, Thus, the series converges if which implies . For the boundary, we have in the case , which diverges since the terms do not go to 0. Similarly, if we get which also diverges since the terms do not go to 0. Therefore, the given series converges absolutely on the interval . Then, to compute the sum for we have # Compute the sum of the series ∑ (-1)n nxn

Find all such that the series converges and compute the sum.

First, we write Then, from Theorem 11.9 we know we can differentiate term-by-term and we have # Compute the sum of the series ∑ nxn

Find all such that the series converges and compute the sum.

First, we write By Theorem 11.9 we know that we can differentiate a power series term-by-term, so we have valid for (since this was where the geometric series was valid).

# Compute the sum of the series ∑ xn / 3n+1

Find all such that the series converges and compute the sum.

From the geometric series expansion we have, This is valid for .