Question

Which of the following statements are true for a divergent series:
(a) The infinite sequence of the partial sums of the series does not have a finite limit.
(b) 2 + 4 + 6 + 8 + ….
(c) Both (a) and (b) are correct.
(d) Only (a) is correct.

Answer 1

Hint: First of all, we will define what is a divergent series and see some examples to understand it better. Then we will check whether the given series complies with our definition or not. And thus, we find the answer to this question.

Complete step-by-step answer:
To understand divergent series better, let us first define convergent series. The definition of divergent series is opposite to that of convergent series.
If a series converges, the individual terms of the series must approach zero. This means, the series must become smaller and smaller as we move forward in the series.
Examples of convergent series are $1+\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{8}+....$
In this series, as we go forward, the number gets more and more divided by 2. So, we can say that the sum of this series approaches 0. In fact, the above series is an infinite Geometric Progression (GP).
So, as the divergent series is opposite to convergent series, in a divergent series, the sum will approach $\infty $ or $-\infty $. Thus, the series does not have a finite limit.
Examples of divergent series are 1 + 2 + 3 + 4 + ….,
Therefore, we can say that a divergent series is an infinite sequence of the partial sums of the series that does not have a finite limit.
Hence, option (a) verifies.
Now, we will see option (b).
The given series is 2 + 4 + 6 + 8 +…..
As we can see, it is a sequence of partial sums. As we move forward, the number becomes bigger and bigger and hence approaches $\infty $.
Thus, we can say that the given series 2 + 4 + 6 + 8… is a divergent series.
Thus, option (b) also verifies.
So, the correct answer is “Option C”.

Note: The conditions for convergent series described above are not compulsorily true for all converging series. i.e. not all series whose sum approaches 0 are converging series. The harmonic series is a diverging series, even though its sum approaches zero.