From the uploaded image, the problem involves evaluating two sums:

1. $S = \sum_{k=1}^n k \cdot \cosh k$
2. $T = \sum_{k=1}^n \frac{k}{\sinh k}$

Let's go through the steps for solving these sums symbolically or computationally, depending on the given $n$.

Solution Strategy

1. For $S$:

   • $\cosh k$ is defined as $\frac{e^k + e^{-k}}{2}$. Substituting this into the sum: $S = \sum_{k=1}^n k \cdot \frac{e^k + e^{-k}}{2} = \frac{1}{2} \sum_{k=1}^n k e^k + \frac{1}{2} \sum_{k=1}^n k e^{-k}.$
   • Separate these into two distinct sums and simplify.

2. For $T$:

   • $\sinh k$ is defined as $\frac{e^k - e^{-k}}{2}$. Substituting into the sum: $T = \sum_{k=1}^n \frac{2k}{e^k - e^{-k}}.$
   • Simplify this further and evaluate for a specific $n$, if needed.

Would you like me to evaluate these sums numerically for a specific value of $n$ or provide a general symbolic expression for these terms? Let me know!

Additional Questions

1. Would you like a detailed breakdown of each step in symbolic computation?
2. Do you need numerical evaluations for a specific value of $n$?
3. Should the sums be graphed to analyze their growth as $n$ increases?
4. Would you like to understand properties of hyperbolic functions used here?
5. Do you want a comparison of these sums with simpler arithmetic or geometric sums?

Tip: When working with sums involving exponential terms, breaking them into geometric series can often simplify evaluations!