You will need

- The textbook on higher mathematics, the table of convergence tests

Instruction

1

By definition a series is called convergent, if there is a finite number, which obviously is more than the sum of the elements of this

**series**. In other words, the series converges if the sum of its elements is finite. To reveal the fact, is the sum of a finite or infinite will help convergence tests for**series**.2

One of the easiest signs of convergence is a sign of convergence Leibniz. We can use, if the series is alternating (that is, each successive member of the

**series**changes the sign from "plus" to "minus"). On the basis of Leibniz alternating series is convergent if the last member**of a number**modulo tends to zero. For this purpose, the limit of the function f(n) flock n to infinity. If this limit is zero, then the series converges, otherwise diverges.3

Another common way to check the number on

**the convergence**.**the convergence**- limiting characteristic of d'alembert. To use it, we divide the n-th term of the sequence to the previous ((n-1)-yy). This relationship we computed, its result is taken modulo (n again to aspire to infinity). If we get a number less than unity, the series converges, otherwise the series diverges.4

Radical sign d'alembert is somewhat similar to the previous one: we extract the root of n-th degree of the n-th member. If we get a number less than unity, then the sequence converges, sum of its members is a finite number.

5

In some cases (when we cannot apply the sign of d'alembert) advantageous to use the integral sign by Cauchy. For this a recorded function

**series**under the integral, differential take n between the limits from zero to innity (this integral is called improper). If the numerical value of this improper integral equals a finite number, the series is convergent.6

Sometimes in order to find out what type of a number, not necessarily to use the convergence tests. You can just compare it with another convergent series. If the number is less than obviously convergent

**series**, then it is also convergent.