To determine whether the series

$\sum_{n=1}^{\infty} \frac{2^n + 3^n}{10^{n/2}}$

is absolutely convergent, conditionally convergent, or divergent, we need to examine the behavior of its general term:

$a_n = \frac{2^n + 3^n}{10^{n/2}}.$

Step 1: Simplify the General Term

Since $2^n$ and $3^n$ are exponential terms, we can approximate the general term by focusing on the dominant term as $n$ grows large. Here, $3^n$ grows faster than $2^n$, so $2^n + 3^n \approx 3^n$ for large $n$. Thus,

$a_n \approx \frac{3^n}{10^{n/2}}.$

We can rewrite $10^{n/2}$ as $(\sqrt{10})^n$, giving:

$a_n \approx \left( \frac{3}{\sqrt{10}} \right)^n.$

Step 2: Check for Convergence of $\sum a_n$

Now, let's examine whether the series converges by considering the ratio of $3$ and $\sqrt{10}$.

1. $\sqrt{10} \approx 3.162$, so $\frac{3}{\sqrt{10}} < 1$.
2. Since $\left( \frac{3}{\sqrt{10}} \right)^n$ is a geometric sequence with a ratio less than 1, it converges.

Step 3: Absolute Convergence

Since we approximated $a_n$ with $\left( \frac{3}{\sqrt{10}} \right)^n$, and this series converges absolutely (as a geometric series with ratio less than 1), we conclude that the original series

$\sum_{n=1}^{\infty} \frac{2^n + 3^n}{10^{n/2}}$

is absolutely convergent.

Would you like a detailed explanation of each approximation step?

Here are five related questions that extend this topic:

1. How would you determine if a series is divergent using the Ratio Test?
2. What is the difference between absolute and conditional convergence?
3. How can you apply the root test for convergence in exponential series?
4. Can you find the sum of this series if it were geometric?
5. What happens if the base of the exponential function changes?

Tip: In convergence tests, focusing on the dominant term often simplifies the analysis for large $n$.