Math Problem Statement

What is the limit of the sum as n approaches infinity for the series: (1/2 + 3/2^2 + 5/2^3 + ... + (2n - 1)/2^n)?

Solution

The given image represents a limit of a sum as nn \to \infty:

limn(12+322+523++2n12n)\lim_{n \to \infty} \left( \frac{1}{2} + \frac{3}{2^2} + \frac{5}{2^3} + \cdots + \frac{2n - 1}{2^n} \right)

This is a series where each term has the general form:

2k12kfork=1,2,3,\frac{2k - 1}{2^k} \quad \text{for} \quad k = 1, 2, 3, \dots

To analyze this series:

  1. The General Term: The general term is 2k12k\frac{2k - 1}{2^k}, and we are interested in summing this for k=1k = 1 to nn, then taking the limit as nn \to \infty.

  2. Splitting the Terms: 2k12k=2k2k12k\frac{2k - 1}{2^k} = \frac{2k}{2^k} - \frac{1}{2^k} This allows us to rewrite the entire sum as two separate sums: k=1n2k12k=2k=1nk2kk=1n12k\sum_{k=1}^n \frac{2k - 1}{2^k} = 2 \sum_{k=1}^n \frac{k}{2^k} - \sum_{k=1}^n \frac{1}{2^k}

  3. Evaluating the Series:

    • The second sum, k=112k\sum_{k=1}^\infty \frac{1}{2^k}, is a geometric series with the sum: k=112k=1\sum_{k=1}^\infty \frac{1}{2^k} = 1

    • The first sum, k=1k2k\sum_{k=1}^\infty \frac{k}{2^k}, is a known series, and its sum is: k=1k2k=2\sum_{k=1}^\infty \frac{k}{2^k} = 2

  4. Putting it Together: limn(2k=1nk2kk=1n12k)=2×21=41=3\lim_{n \to \infty} \left( 2 \sum_{k=1}^n \frac{k}{2^k} - \sum_{k=1}^n \frac{1}{2^k} \right) = 2 \times 2 - 1 = 4 - 1 = 3

Thus, the value of the limit is:

3\boxed{3}

Would you like further details on any step or have any questions? Here are some related topics:

  1. What are the properties of geometric series?
  2. How do you derive the sum of krk\sum \frac{k}{r^k}?
  3. How can series be used to approximate functions?
  4. What is the relationship between finite sums and limits?
  5. How does this relate to calculus in infinite series?

Tip: Whenever you split a series, try to evaluate each part separately to simplify the overall problem!

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Math Problem Analysis

Mathematical Concepts

Series and Sequences
Infinite Series
Geometric Series
Limits

Formulas

General term: (2k - 1)/2^k for k = 1, 2, 3,...
Splitting terms: (2k - 1)/2^k = 2k/2^k - 1/2^k
Sum of infinite geometric series: sum(1/2^k) = 1
Known series: sum(k/2^k) = 2

Theorems

Geometric Series Theorem
Limit Theorems for Series

Suitable Grade Level

Undergraduate Calculus or Advanced High School Calculus