Question

Evaluate each series. $$\sum_{i=1}^{3}\left(i^{3}-i\right)$$

   Evaluate each series.
$$\sum_{i=1}^{3}\left(i^{3}-i\right)$$

Precalculus

Margaret L. Lial,… 6th Edition

Chapter 11, Problem 53

Instant Answer

Solved by Expert Rakesh Kumar Sharma

08/27/2020

Step 1

We have $i^3 - i$. When $i = 1$, we have $1^3 - 1 = 1 - 1 = 0$. When $i = 2$, we have $2^3 - 2 = 8 - 2 = 6$. When $i = 3$, we have $3^3 - 3 = 27 - 3 = 24$. Now, let's add up these values: $0 + 6 + 24 = 30$. Show more…

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Evaluate each series. $$\sum_{i=1}^{3}\left(i^{3}-i\right)$$

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Key Concepts

Summation Notation

Summation notation uses the sigma (?) symbol to represent the sum of a sequence of terms indexed by a variable. It defines the range of the index and encapsulates a repeated addition of an expression evaluated at different integer values within that range.

Finite Series Evaluation

Finite series evaluation involves computing the sum of a sequence when the number of terms is limited. This includes substituting each integer index into the series' general term and adding the results, which is a fundamental technique in algebra and calculus.

Polynomial Expressions in Series

Polynomial expressions in series consist of terms where each is defined by a polynomial function of the index variable. Evaluating such series requires calculating the polynomial's value for each index and then summing those values, often employing strategies that simplify the computation.

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