Math Problem Statement

Calcula el valor de S = 1/3 + 2/3 + 2/9 + 1/6 + 4/27 + 1/24 + ...

Solution

The series in the image is given as:

S=13+23+29+16+427+124+S = \frac{1}{3} + \frac{2}{3} + \frac{2}{9} + \frac{1}{6} + \frac{4}{27} + \frac{1}{24} + \dots

Let’s calculate the sum step by step to determine the value of SS.

Step 1: Simplify the first few terms of the series

We will calculate the partial sums of the first few terms.

S1=13=0.333S_1 = \frac{1}{3} = 0.333 S2=13+23=1S_2 = \frac{1}{3} + \frac{2}{3} = 1 S3=13+23+29=1+29=1+0.222=1.222S_3 = \frac{1}{3} + \frac{2}{3} + \frac{2}{9} = 1 + \frac{2}{9} = 1 + 0.222 = 1.222 S4=1.222+16=1.222+0.167=1.389S_4 = 1.222 + \frac{1}{6} = 1.222 + 0.167 = 1.389 S5=1.389+427=1.389+0.148=1.537S_5 = 1.389 + \frac{4}{27} = 1.389 + 0.148 = 1.537 S6=1.537+124=1.537+0.042=1.579S_6 = 1.537 + \frac{1}{24} = 1.537 + 0.042 = 1.579

Step 2: Compare with answer choices

The value of SS after summing the first six terms is approximately 1.579, which is very close to 1491.556\frac{14}{9} \approx 1.556.

Thus, the correct answer is:

D.149\boxed{D. \frac{14}{9}}


Would you like further details on this calculation, or have any additional questions?

Here are 5 related questions for further exploration:

  1. What is the general method for summing an infinite series?
  2. How can we determine if a series converges or diverges?
  3. What other types of series are commonly encountered in calculus?
  4. How can we approximate an infinite series sum?
  5. What are some applications of series in real-life problems?

Tip: When dealing with series, always check whether it converges or diverges before attempting to find the sum.

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Math Problem Analysis

Mathematical Concepts

Series and Sequences
Fraction Addition

Formulas

Sum of Infinite Series Formula
Addition of Fractions

Theorems

Convergence of Series

Suitable Grade Level

Grades 10-12