Looking for an in depth proof to this problem if possible. Must be clear with partial sums ### Series Convergence or Divergence AnalysisDetermine whether each of the following series converges or diverges. If the series converges, give its sum. Provide complete and careful proofs of your answers.\[\sum_{k=1}^{\infty} a_k, \text{ where } a_k = \begin{cases} k^2 & \text{if } 1 \leq k \leq 10 \\0 & \text{if } k > 10 \end{cases}\](Be clear about the sequence of partial sums.)### Explanation:To analyze the convergence or divergence of the given series, we consider the definition of $a_k$ provided.1. For $1 \leq k \leq 10$: \[ a_k = k^2 \]2. For $k > 10$: \[ a_k = 0 \]### Sequence of Partial Sums:To find the partial sums $S_n$ of the series, sum the terms from $k=1$ to $k=n$:\[S_n = \sum_{k=1}^{n} a_k\]Given $a_k$ is non-zero only for $1 \leq k \leq 10$, the partial sum for $n \geq 10$ becomes:\[S_n = \sum_{k=1}^{10} k^2\]Calculating the sum for $k$ from 1 to 10:- When $k = 1$: $1^2 = 1$- When $k = 2$: $2^2 = 4$- When $k = 3$: $3^2 = 9$- When $k = 4$: $4^2 = 16$- When $k = 5$: $5^2 = 25$- When $k = 6$: $6^2 = 36$- When $k = 7$: $7^2 = 49$- When $k = 8$: $8^2 = 64$- When $k = 9$: $9^2 = 81$- When \(k =

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Description: Looking for an in depth proof to this problem if possible. Must be clear with partial sums ### Series Convergence or Divergence AnalysisDetermine whether each of the following series converges or diverges. If the series converges, give its sum. Provi

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Determine whether each of the following series converges or diverges. If the series converges, give its sum. Give complete and careful proofs of your answers. Se if1sks 10 10 ifk> 10. a, where a (Be clear about the sequence of partial sums.)

Determine whether each of the following series converges or diverges. If the series converges, give its sum. Give complete and careful proofs of your answers. Se if1sks 10 10 ifk> 10. a, where a (Be clear about the sequence of partial sums.)

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.

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Looking for an in depth proof to this problem if possible. Must be clear with partial sums

### Series Convergence or Divergence Analysis

Determine whether each of the following series converges or diverges. If the series converges, give its sum. Provide complete and careful proofs of your answers.

\[
\sum_{k=1}^{\infty} a_k, \text{ where } a_k =
\begin{cases}
k^2 & \text{if } 1 \leq k \leq 10 \\
0 & \text{if } k > 10
\end{cases}
\]

(Be clear about the sequence of partial sums.)

### Explanation:
To analyze the convergence or divergence of the given series, we consider the definition of $a_k$ provided.

1. For $1 \leq k \leq 10$:
\[
a_k = k^2
\]

2. For $k > 10$:
\[
a_k = 0
\]

### Sequence of Partial Sums:
To find the partial sums $S_n$ of the series, sum the terms from $k=1$ to $k=n$:
\[
S_n = \sum_{k=1}^{n} a_k
\]

Given $a_k$ is non-zero only for $1 \leq k \leq 10$, the partial sum for $n \geq 10$ becomes:
\[
S_n = \sum_{k=1}^{10} k^2
\]

Calculating the sum for $k$ from 1 to 10:
- When $k = 1$: $1^2 = 1$
- When $k = 2$: $2^2 = 4$
- When $k = 3$: $3^2 = 9$
- When $k = 4$: $4^2 = 16$
- When $k = 5$: $5^2 = 25$
- When $k = 6$: $6^2 = 36$
- When $k = 7$: $7^2 = 49$
- When $k = 8$: $8^2 = 64$
- When $k = 9$: $9^2 = 81$
- When \(k =

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