Answer
a) Convergent
b) The limit: $2$
Work Step by Step
a) We will use the theorem which states that if a sequence ${a_n}$ is bounded and monotonic, then it converges.
To prove that the sequence is bounded and monotone:
Boundedness:
To check if the sequence is bounded, we need to find upper and lower bounds. Notice that for all $n$, $a_n$ is greater than or equal to $2$ (because we're adding a positive number to $2$), so the sequence is bounded from below by $2$. Additionally, since $0 < \frac{1}{5^n} \leq 1$ (for all $n$), we can say that $a_n$ is less than or equal to $2 + 1 = 3$ for all $n$. Therefore, the sequence ${a_n}$ is bounded above by $3$.
Monotonicity:
We need to determine if the sequence is monotonic, which means it's either increasing or decreasing indefinitely. In this case, the sequence is clearly decreasing because as $n$ increases, the term $\frac{1}{5^n}$ decreases, and we're adding it to $2$. Therefore, the sequence ${a_n}$ is monotonically decreasing.
b) As shown from the table, the graph is approaching $2$, so the limit is $2$.
