Question

How do you find if the sequence converges?

Answer 1

Hint: Now to find if a sequence is converging or diverging we need to check the end behavior of a sequence. Hence ${{a}_{n}}$ is general term of sequence then we need to first find the value of $\underset{n\to \infty }{\mathop{\lim }}\,{{a}_{n}}$ If the value is finite then the sequence is convergent

Complete step by step solution:
Now let us first understand what sequences are.
Sequences are nothing but an infinite list of numbers.
Hence we have first term, second term and so on…
Now in general we say ${{a}_{1}}$ is first term of sequence, ${{a}_{2}}$ is second term and so on we have ${{a}_{n}}$ is the last term.
Now let us check an example of a sequence.
Let us say ${{n}^{th}}$ term of a sequence is given by ${{a}_{n}}=\dfrac{1}{n}$
Hence such sequence is given by $1,\dfrac{1}{2},\dfrac{1}{3},\dfrac{1}{4},\dfrac{1}{5},...$
Hence if we have a general term of sequence then we can write the whole sequence by substituting values of n as natural numbers.
Now a sequence can be converging or diverging.
If the end term of sequence is finite then the sequence is convergent.
Hence if $\underset{n\to \infty }{\mathop{\lim }}\,{{a}_{n}}=l$ where l is any finite number then we say that the limit of sequence exist and the sequence is converging. If the limit is not finite then we have the limit does not exist and hence the sequence is diverging.

Note: Now note that the sequence is just a list of numbers. While series is nothing but addition of the terms of sequence. Now note that for a converging sequence we have the distance between the two consecutive terms get closer or is constant after a period of time. But for diverging sequences the distance between consecutive terms increases.