Reasoning as in the given problem, what is the value of0.3+0.03+0.003+...?

rocedwrp 2021-02-03 Answered

Reasoning as in the given problem, what is the value of
0.3+0.03+0.003+...?
Working with series Consider the infinite series
0.9+0.09+0.009+0.0009+...,
where each term of the sum is 110 of the previous term.
a. Find the sum of the first one, two, three, and four terms of the series.
b. What value would you assign to the infinite series 0.9+0.09+0.009+ ⋅ ⋅ ⋅?

You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Expert Answer

Latisha Oneil
Answered 2021-02-04 Author has 100 answers

Sum of infinite geometric series:
Any series in the form of a,ar,ar2,... is called infinite geometric series and sum of infinite geometry series is,
S=a1r
Where a is the first term and r is a common ratio.
And Sum of geometric series:
Sn=a(1rn)1r
Given series,
0.3+0.03+0.003+
Since
0.3=0.3
0.03=0.3(0.1)
0.003=0.3(0.1)2
So given series is a geometric series with first term a=0.3 and common ratio is 0.1.
So sum of infinite geometry series is,
S=0.310.1
S=0.30.9
S=13
S=0.333...
Hence
0.3+0.03+0.003+...=0.333...
Given series,
0.9+0.09+0.009+0.0009+...
Since
0.9=0.9
0.09=0.9(0.1)
0.009=0.9(0.1)2
So given series is a geometric series with first term a=0.9 and common ratio is 0.1.
a) Evaluate the sum of the first one, two, three, and four terms of the series.
Since sum of geometric series:
Sn=a(1rn)1r
Since a=0.9,r=0.1
Sum of the first term,
S1=0.9(1(0.1)1)1(0.1)=0.9(0.9)0.9=0.9
Sum of the first two terms,
S2=0.9(1(0.1)2)1(0.1)=0.9(0.99)0.9=0.99
Sum of the first three terms,
S3=0.9(1(0.1)3)1(0.1)=0.9(0.999)0.9=0.999
Sum of the first four terms,
S4=0.9(1(0.1)4)1(0.1)=0.9(0.9999)0.9=0.9999
Hence
Sum of the first term =0.9
Sum of the first two terms =0.99
Sum of the first three terms =0.999
Sum of the first four terms =0.9999
b) So sum of infinite geometry series is,
S=0.910.1
S=0.90.9
S=1
Hence
0.9+0.09+0.009+0.0009+...=1

Not exactly what you’re looking for?
Ask My Question
Jeffrey Jordon
Answered 2021-12-16 Author has 2313 answers

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

New questions