# Sequence of Partial Sums

The sequence of partial sums of a given series approaches a fixed value. What can you conclude about this series as a result regarding convergence?

So I know that this type of series is converging but I would like to use an example but am not sure what format to put it in. I am new to calculus and am having to learn it at home because of covid- 19 and am still a bit confused.

$$\sum_{i=0}^∞ 0.48(0.01)^i$$ = 0.4848(repeating)

Would this work as an example and does its sum exist? PLEASE! correct me if I am incorrect in any of this I am just trying my best to understand.

• Its a standard series, the geometric progression. Look up in wikipedia! – jeea Apr 1 '20 at 18:47

Note that $$\sum_{i=0}^\infty a_i$$ is defined as $$\sum_{i=0}^\infty a_i:=\lim_{n\to \infty} \sum_{i=0}^n a_i=\lim_{n\to \infty} S_n$$ where $$S_n$$ is the $$n$$'th partial sum. So if the partial sums converge to a value, then this value is (by definition) $$\sum_{i=0}^\infty a_i$$. In this case, we say the series converges. If the partial sums do not converge to a fixed quantity, then we say the series diverges.