Question

Is the power series ∑ₖ₌₀[infinity] 1/(2k)! (X-10)ᵏ

convergent? If so, what is the radius of convergence?

Answer 1

The power series ∑ₖ₌[infinity] 1/(2k)! (X-10)ⁿ converges for all X, as demonstrated by the ratio test, which indicates absolute convergence. The radius of convergence for this series is therefore infinity.

Step-by-step explanation:

The power series ∑ₖ₌[infinity] 1/(2k)! (X-10)ⁿ is indeed convergent, and we can determine this by using the ratio test for convergence which is often applied to power series. The ratio test states that for the series Σan, if the limit as n approaches infinity of |an+1/an| is less than 1, the series converges absolutely. Applying the ratio test to our series:

1. Calculate |an+1/an| for our series, where an = 1/(2n)! (X-10)n.
2. You will find that as n approaches infinity, |an+1/an| will approach 0 for all X, which is less than 1.
3. This indicates that the series is absolutely convergent for all X, which means the radius of convergence is infinity.

Therefore, this power series behaves like the expansion of the exponential function ex and converges for all real numbers X.