Find general term of a sequence

Question

Help to find a formula for the general term $x_n$ of the sequence and find out whether it is covergent or divergent:

3/2, 3/4 +0.1, 3/6, 3/8+0.1...

Answer 1

If you only had the sequence:

$$ \frac{3}{2},\frac{3}{4},... $$

You would get $a_n$:

$$ a_n=\frac{3}{2n} $$

Assuming the sequence $a_n$ is defined for $n\geq1$:

Now notice that for even values of $n$ you have an additional $0.1$, let's name it $b_n$:

$$ b_{n}=0.1\quad n\quad even\\ b_n=0\quad n\quad odd $$

And for all $n\geq 1$ you get:

$$ b_n=0.1\frac{(1+{(-1)}^{n})}{2} $$

So your sequence is:

$$ x_n=a_n+b_n $$

Or

$$ x_n=\frac{3}{2n}+0.1\frac{(1+{(-1)}^{n})}{2} $$

Clearly $x_n$ has no limit as the limit ${lim}_{n\rightarrow \infty}{\frac{1+{(-1)}^{n}}{2}}$ doesnt exist