Studdy AI Logo
Math  /  Algebra

QuestionEvaluate the infinite geometric series: 1+1/5+1/25+…1+1 / 5+1 / 25+\ldots
1 2 3
65/27 5/4 9/59 / 5 5/6

Studdy Solution

STEP 1

1. The series is an infinite geometric series.
2. The first term of the series is a=1 a = 1 .
3. The common ratio r r can be determined from the series.

STEP 2

1. Identify the first term and the common ratio.
2. Use the formula for the sum of an infinite geometric series.
3. Calculate the sum.

STEP 3

Identify the first term a a and the common ratio r r .
The first term a a is the first term of the series:
a=1 a = 1
The common ratio r r is the ratio of the second term to the first term:
r=1/51=15 r = \frac{1/5}{1} = \frac{1}{5}

STEP 4

Use the formula for the sum S S of an infinite geometric series, which is valid when ∣r∣<1 |r| < 1 :
S=a1−r S = \frac{a}{1 - r}
Substitute the values of a a and r r into the formula:
S=11−15 S = \frac{1}{1 - \frac{1}{5}}

STEP 5

Simplify the expression for the sum:
S=11−15=145 S = \frac{1}{1 - \frac{1}{5}} = \frac{1}{\frac{4}{5}}
S=1×54=54 S = \frac{1 \times 5}{4} = \frac{5}{4}
The sum of the infinite geometric series is:
54 \boxed{\frac{5}{4}}

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord