The given series is:

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

We are asked to determine whether this series is convergent or divergent.

Step 1: Ratio Test

The ratio test is a common method used to determine the convergence of a series. We will apply it here. The ratio test states that for a series $\sum a_k$, if the limit

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

exists and:

• If $L < 1$, the series converges absolutely.
• If $L > 1$, the series diverges.
• If $L = 1$, the test is inconclusive.

Let $a_k = \frac{2^k}{(2k)!}$.

Now, compute the ratio:

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

Simplifying this expression:

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

Step 2: Take the limit as $k \to \infty$

As $k$ becomes very large, the term $(2k+2)(2k+1)$ grows quadratically, so:

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

Since $L = 0 < 1$, by the ratio test, the series converges.

Conclusion:

The series $\sum_{k=0}^{\infty} \frac{2^k}{(2k)!}$ is convergent.

Would you like more details or have any questions?

Here are 5 related questions to deepen your understanding:

1. How does the ratio test help determine the convergence of a series?
2. What is the significance of factorials in determining the behavior of series?
3. Can you apply the root test to the same series, and what would be the result?
4. How does the series compare to a geometric series?
5. What other types of convergence tests could apply to this problem?

Tip: Always simplify factorial expressions carefully when applying the ratio test to avoid algebraic mistakes.