Math Problem Statement

What is the limit of the series \( \lim_{n \to \infty} \left( \frac{3}{5} + \frac{3}{25} - \cdots + 3 \cdot \frac{(-1)^n}{5^n} \right)?

Solution

The problem involves finding the limit of a series as nn \to \infty. Let's analyze and simplify the series.

Problem Breakdown:

The given series is:

limn(35+325+3(1)n5n).\lim_{n \to \infty} \left( \frac{3}{5} + \frac{3}{25} - \cdots + 3 \cdot \frac{(-1)^n}{5^n} \right).

This is a geometric series with the general term:

an=3(1)n5n.a_n = 3 \cdot \frac{(-1)^n}{5^n}.

The first term is 35\frac{3}{5}, and the common ratio is r=15r = -\frac{1}{5}.

Infinite Geometric Series Formula:

For an infinite geometric series with first term aa and common ratio rr, where r<1|r| < 1, the sum is given by:

S=a1r.S = \frac{a}{1 - r}.

Applying to the Series:

  1. First term: a=35a = \frac{3}{5},
  2. Common ratio: r=15r = -\frac{1}{5}.

Substituting into the formula:

S=351(15)=351+15=3565=36=12.S = \frac{\frac{3}{5}}{1 - \left(-\frac{1}{5}\right)} = \frac{\frac{3}{5}}{1 + \frac{1}{5}} = \frac{\frac{3}{5}}{\frac{6}{5}} = \frac{3}{6} = \frac{1}{2}.

Final Answer:

limn(35+325+3(1)n5n)=12.\lim_{n \to \infty} \left( \frac{3}{5} + \frac{3}{25} - \cdots + 3 \cdot \frac{(-1)^n}{5^n} \right) = \frac{1}{2}.

Would you like further details on the steps or explanation?


Here are 5 related questions for deeper exploration:

  1. How does the sum of a finite geometric series differ from an infinite geometric series?
  2. What conditions must hold for an infinite geometric series to converge?
  3. Can you find the sum of the series if the common ratio rr is greater than or equal to 1?
  4. How do alternating signs in the terms (due to (1)n(-1)^n) affect the convergence of a series?
  5. How would this series change if the first term were different?

Tip: When working with infinite series, always check the magnitude of the common ratio r|r| to determine convergence.

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Math Problem Analysis

Mathematical Concepts

Infinite Series
Geometric Series
Convergence

Formulas

Sum of an infinite geometric series: S = a / (1 - r)

Theorems

Convergence of Infinite Geometric Series

Suitable Grade Level

Grades 11-12