To determine the interval of convergence for the series:

$\sum_{k=0}^{\infty} \frac{(-1)^k x^k}{(2k)!}$

We can use the Ratio Test for convergence. The Ratio Test states that for a series $\sum a_k$, the series converges absolutely if:

$L = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| < 1$

If $L > 1$, the series diverges. If $L = 1$, the test is inconclusive, and we need to investigate further.

Step-by-Step Solution

1. General Term: The general term in the series is:

   $a_k = \frac{(-1)^k x^k}{(2k)!}$

2. Applying the Ratio Test: We calculate the limit of the ratio of successive terms:

   $\frac{a_{k+1}}{a_k} = \frac{(-1)^{k+1} x^{k+1}}{(2(k+1))!} \cdot \frac{(2k)!}{(-1)^k x^k}$

   Simplifying this expression:

   $\frac{a_{k+1}}{a_k} = \frac{(-1)^{k+1}}{(-1)^k} \cdot \frac{x^{k+1}}{x^k} \cdot \frac{(2k)!}{(2k+2)!}$

   $\frac{a_{k+1}}{a_k} = (-1) \cdot x \cdot \frac{1}{(2k+1)(2k+2)}$

   Thus,

   $\left| \frac{a_{k+1}}{a_k} \right| = \frac{|x|}{(2k+1)(2k+2)}$

3. Taking the Limit: As $k \to \infty$, the denominator $(2k+1)(2k+2)$ grows without bound, so:

   $\lim_{k \to \infty} \frac{|x|}{(2k+1)(2k+2)} = 0$

4. Conclusion from Ratio Test: Since the limit is $0$ for all values of $x$, the Ratio Test tells us that the series converges for all $x$.

Interval of Convergence:

The series converges for all real values of $x$, so the interval of convergence is:

$(-\infty, \infty)$

Additional Notes:

• The series converges absolutely for all $x$, as shown by the Ratio Test.
• This result is similar to the series for the exponential function, though this series involves only even powers of $x$.

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Would you like more details on this approach? Or do you have other questions?

Here are some related questions to explore further:

1. How does the Ratio Test apply to other series like $\sum \frac{x^n}{n!}$?
2. What are the conditions for applying the Root Test for series convergence?
3. How can you determine the radius of convergence for a power series?
4. What are the implications of convergence at the boundary points for series?
5. Can we apply the alternating series test to determine convergence of this series?

Tip: When using the Ratio Test, if the limit approaches zero, the series always converges for any value of $x$.