**Definition: **If {a_{no}} is a sequence, so the infinite sum:

The_{1} + a_{2} + a_{3} +… + A_{no} +… =

it's called **series**.

Each number **The _{i}** is a term in the series;

**The _{no}** is the generic term n.

To define the sum of infinite installments, we consider the** partial sums**

s_{1 }= a_{1}

s_{2} = a_{1} + a_{2}

s_{3} = a_{1} + a_{2} + a_{3}

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s_{no} = a_{1} + a_{2} + a_{3} +… + A_{n-1} + a_{no}

And the** sequence of partial sums**

s_{1}, S_{2}, S_{3},… , S_{no},…

If this sequence has limit S, then the series **converge** and your sum is **S.**

That is: If , then the series converges and its sum is the_{1}+ a_{2}+ a_{3}+… + A_{no}… = S

If the sequence {S_{no}} has no limit, so the series **differ**.

If the series converge then .

**Note: *** The reciprocal of this theorem is false, that is, there are series whose generic term tends to zero and which are not convergent.

* It is worth the counter: "*if the limit is not zero then the series no converge*", which constitutes the following test.

### Divergence Test

Given the series ,

**diverges.**