A series of a sequence is the sum of all the elements in that sequence. If we want to consider whether a series is convergent or divergent, the most primary factor is to consider the “speed” of convergence. Thant is, how fast for the next term to get smaller. Let’s have the most famous series to be an example: The harmonic series. As we all know that the harmonic series is divergent, even though the sequence of it is getting smaller. However, if we can make it to become 1/(n)^1.1 in order to have a faster decreasing in the next elements, the series is going to be convergent slowly.

Think it like this situation:

When you are using a sharp scissors to cut through a paper, and you just want a smooth edge as possible as you can, no matter if it is a straight line. Then, the best way to do is to cut it very quickly. However, you probably will get a ragged cut if you do it slowly.

A sequence “an” is a list of elements that have its own order to follow with an increasing in positive integer number n. If we want to consider a sequence is convergent or divergent, we need to look at the tendency of the elements as n goes to infinity.

As long as the elements is getting closer and closer to one specific number, while n is approaching to infinity, then we can say this sequence is convergent. Otherwise, the sequences are divergent.

For example, a sequence ((-1)^n)/n has a list of elements {-1, 1/2, -1/3, 1/4, -1/5…}. We can see that although elements are jumping back and forth between the x-axis, as n gets as large as we need, this sequence is as close as we want to 0. That is, the sequence is convergent.

For a divergent sequence example, we can have a sequence (-1)^n, which has a list of elements {-1, 1, -1, 1, -1, 1…}. Observably, it is not going to get close to one specific number. So, this sequence is divergent.

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