**Mathematics Exercise: Summation and Index Variable Transformation**### IV. Problems on Summation Notation**(a)** Write the following series using summation notation:\[ 1 + \frac{1}{3} + \frac{1}{3^2} + \cdots + \frac{1}{3^n} \]---**(b)** Transform the summation by changing the index variable from $ k $ to $ j $ using the given relationship $ j = k-1 $:\[\sum_{k=1}^{7} k(k+2)(k-3)\]---- **Task:** Write out the first four terms of the following summation: \[ \sum_{j=1}^{n} j(j-1) \] **Include at least 4 terms:**---**(d)** Compute the product:\[\prod_{j=1}^{4} (-2)^j = \]

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Description: **Mathematics Exercise: Summation and Index Variable Transformation**### IV. Problems on Summation Notation**(a)** Write the following series using summation notation:\[ 1 + \frac{1}{3} + \frac{1}{3^2} + \cdots + \frac{1}{3^n} \]---**(b)** Transform

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(a) Write using summation notation: 1 +1 + 1 + 32 + 1 3n (b) Transform the summation by making the specified change of index variable k to variable j using the relationship j = k-1: 7 E k(k+2)(k-3) k-1 Write out the first four terms of the summation below.: Include at least 4 terms j G-1) = j=1 (d) Compute the product (-2) = j = 1

(a) Write using summation notation: 1 +1 + 1 + 32 + 1 3n (b) Transform the summation by making the specified change of index variable k to variable j using the relationship j = k-1: 7 E k(k+2)(k-3) k-1 Write out the first four terms of the summation below.: Include at least 4 terms j G-1) = j=1 (d) Compute the product (-2) = j = 1

Advanced Engineering Mathematics

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Chapter2: Second-order Linear Odes

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Concept explainers

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence. It was invented by German mathematician Carl Friedrich Gauss.

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Question

**Mathematics Exercise: Summation and Index Variable Transformation**

### IV. Problems on Summation Notation

**(a)** Write the following series using summation notation:
\[ 1 + \frac{1}{3} + \frac{1}{3^2} + \cdots + \frac{1}{3^n} \]

---

**(b)** Transform the summation by changing the index variable from $ k $ to $ j $ using the given relationship $ j = k-1 $:

\[
\sum_{k=1}^{7} k(k+2)(k-3)
\]

---

- **Task:** Write out the first four terms of the following summation:

\[
\sum_{j=1}^{n} j(j-1)
\]

**Include at least 4 terms:**

---

**(d)** Compute the product:

\[
\prod_{j=1}^{4} (-2)^j =
\]

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