Question

How do you find the sum of the convergent series $0.1 + 0.01 + 0.001 + ...?$ if the convergent series is not convergent, how do you know?

Answer 1

Hint: As we know that a series is defined as an expression in which infinitely many terms are added one after the another to a given starting quantity. In this question we are given a GP series or Geometric progression series. It is a sequence of non zero numbers where each term after the first is found by multiplying the previous one by a fixed number called the common ratio. We will first find the common ratio of the given series and then solve it on the basis of that.

Complete step by step solution:
In the given question we have $0.1 + 0.01 + 0.001 + ...$.
We know that the geometric series having common ratio $r < 1$ is called a Convergent series while the geometric ratio having $r > 1$ is called a Divergent series.
We can find the common ratio by the formula $r = \dfrac{{{a_n}}}{{{a_n} - 1}}$.
So it gives us $r = \dfrac{{0.01}}{{0.01 - 1}} = \dfrac{{0.01}}{{0.1}} = 0.1$.
Hence the given series is Convergent geometric series.
We know that the sum of convergent series is
${S_n} = \dfrac{{{a_1}}}{{1 - r}}$, where ${a_1} = 0.1$.
Now by putting the values we have $\dfrac{{0.1}}{{1 - 0.1}} = \dfrac{{0.1}}{{0.9}}$.
Hence the given series is convergent and the sum i.e. ${S_n} = \dfrac{1}{9}$.
So, the correct answer is “${S_n} = \dfrac{1}{9}$”.

Note: When we get further and further in a sequence the terms get closer and closer to a specified limit, this signifies the convergence of series.. Before solving this type of question we should be fully aware of the convergent and divergent series. If the value of common ratio i.e. $r = 1$, then it is inconclusive as the limit of this falls to exist