Question

Evaluate. $$\sum_{i=0}^{3}\left(\frac{1}{2}\right)^{2 k}$$

   Evaluate.
$$\sum_{i=0}^{3}\left(\frac{1}{2}\right)^{2 k}$$

Calculus: One and Several Variables

Calculus: One and Several Variables

Saturnino L. Salas,… 10th Edition

Chapter 12, Problem 9

Instant Answer

Solved by Expert Nick Johnson

11/09/2021

Step 1

We can write out the terms of the sum as follows: $$\left(\frac{1}{2}\right)^{2 \cdot 0} + \left(\frac{1}{2}\right)^{2 \cdot 1} + \left(\frac{1}{2}\right)^{2 \cdot 2} + \left(\frac{1}{2}\right)^{2 \cdot 3}$$ Show more…

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Evaluate. $$\sum_{i=0}^{3}\left(\frac{1}{2}\right)^{2 k}$$

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Key Concepts

Summation Notation

Summation notation is a concise way of representing the addition of a sequence of terms defined by an index variable. It allows for compact representation of long sums and is fundamental in many areas of mathematics, especially in series and sequence analysis.

Geometric Series

A geometric series is a series where each term after the first is obtained by multiplying the previous term by a constant, known as the common ratio. Recognizing a sequence as a geometric series enables the use of specialized formulas to find its sum, particularly when the series is finite or convergent.

Exponentiation Rules

Exponentiation rules govern the operations involving powers and exponents. They allow us to simplify expressions where exponents are multiplied or raised to another power. In this context, rewriting an expression like (1/2) raised to a multiple of an index helps in revealing the structure of the series as geometric.

Finite Geometric Series Formula

The finite geometric series formula provides a way to calculate the sum of a fixed number of terms in a geometric sequence without having to add each term individually. It is especially useful for series with a non-unit common ratio and when the series is short, as it yields a closed-form expression.

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