Question

What are two examples of convergent sequences?

Answer 1

Hint: We solve this problem by using the definition of convergence and some tests used to find the convergence. There are mainly 2 tests we use for writing the examples of convergent sequences.
(1) p – series test:
If the sequence in the form $\sum{\dfrac{1}{{{n}^{p}}}}$ then,
If $p<1$ then the sequence converges and if $p\ge 1$ then the sequence diverges.
(2) Root test:
If the ${{n}^{th}}$ term of a sequence is ${{a}_{n}}$ and $l=\underset{n\to \infty }{\mathop{\lim }}\,\sqrt[n]{\left| {{a}_{n}} \right|}$
If $l < 1$ then the sequence converges and if $l\ge 1$ then the sequence diverges.

Complete step by step answer:
We are asked to give two examples of convergent sequences.
Let us use two tests for convergence to find the examples of convergent sequences.
First let us use the p – series test.
We know that the p – series test is given as,
If the sequence in the form $\sum{\dfrac{1}{{{n}^{p}}}}$ then,
If $p < 1$ then the sequence converges and if $p\ge 1$ then the sequence diverges.
Now, let us take an example where $p < 1$ in the sequence $\sum{\dfrac{1}{{{n}^{p}}}}$ then we get one example as $\sum{\dfrac{1}{\sqrt{n}}}$
Here, we can see that in $\sum{\dfrac{1}{\sqrt{n}}}$ the value of $'p'$ is $\dfrac{1}{2}$ which is less than 1.
Now, let us use the root test.
We know that the root test is given as,
If the ${{n}^{th}}$ term of a sequence is ${{a}_{n}}$ and $l=\underset{n\to \infty }{\mathop{\lim }}\,\sqrt[n]{\left| {{a}_{n}} \right|}$
If $l < 1$ then the sequence converges and if $l\ge 1$ then the sequence diverges.
Let us assume that the ${{n}^{th}}$ term of sequence as,
$\Rightarrow {{a}_{n}}=\dfrac{1}{{{3}^{n}}}$
Now, by using the root test to above sequence then we get,
$\begin{align}
  & \Rightarrow l=\underset{n\to \infty }{\mathop{\lim }}\,\sqrt[n]{\dfrac{1}{{{3}^{n}}}} \\
 & \Rightarrow l=\underset{n\to \infty }{\mathop{\lim }}\,\dfrac{1}{3}=\dfrac{1}{3} \\
\end{align}$
Here, we can see that the value of $'l'$ is less than 1 so that the sequence is convergent.
Therefore, we can conclude that the two examples of convergent sequence are given as,
(1) $\sum{\dfrac{1}{\sqrt{n}}}$
(2) $\sum{\dfrac{1}{{{3}^{n}}}}$

Note: The main mistake is done in p – series test. The p – series test have the summation as $\sum{\dfrac{1}{{{n}^{p}}}}$ then we can compare the value of $'p'$ with respect to ‘1’ to get the nature of sequence.
But some students may take the p – series as $\sum{{{n}^{p}}}$ and compare the value of $'p'$ which gives the wrong answer.