# 1. Series

### Series

Given a sequence of real numbers , we can consider a series , simply as a sequence of partial sums:

The sum of a given series is the limit of sequence , if it exists. If it is finite, then we say that the series converges.

E.g. series is convergent and sums up to . Indeed, , which converges to .

### Necessary condition

It is easy to notice that if a series is convergent then the sequence is convergent to zero. In particular, if is not convergent or converges to a non-zero limit, then cannot be convergent.

E.g. series is not convergent, because sequence is not convergent. Also is not convergent, because converges to .

Notice that the reverse implication is not true. A series may not be convergent even if converges to zero. E.g. sequence .

### Arithmetic and comparison criterion

Since series can be seen as sequences, many theorems on arithmetic of limits make sense also in the case of series. For example, limit of sum of two series is the sum of their sums. Notice that multiplying is a bit tricky. Multiplication of elements of series is not the same as multiplication of partial sums.

But we have a criterion which is implied by the three sequences theorem. If for all greater then some number and is convergent then also converges.

E.g. series is obviously convergent, because .

### D’Alembert criterion

D’Alembert criterion states what can be said how convergence of depending on limit of . If this limit is , the series is convergent, but if , it is not.

E.g. is convergent, because .

Notice that the criterion does not give any result if the limit equals .

### Cauchy criterion

Cauchy criterion states what can be said how convergence of depending on limit of . If this limit is , the series is convergent, but if , it is not.

E.g. is convergent, because .

Notice that the criterion does not give any result if the limit equals .

### Leibniz theorem

Leibniz theorem states that if is non-increasing and converges to zero, then is convergent.

In particular, is convergent, since is non-increasing and converges to zero.

### Absolute convergence

We shall say that is absolutely convergent, if is convergent. Notice that, if a series is absolutely convergent, then it is convergent.

E.g. is convergent, but is not absolutely convergent. On the other hand, is absolutely convergent (and also obviously convergent).