Is there any way to get convergent series if I know the sum? If I am given the limit of a convergent series is there any way that I can find the series?
Is it possible that for any given limit there are infinitely many or no solutions at all?
 A: Have a look at the Wikipedia on the Riemann Series theorem:
In mathematics, the Riemann series theorem (also called the Riemann rearrangement theorem), named after 19th-century German mathematician Bernhard Riemann, says that if an infinite series is conditionally convergent, then its terms can be arranged in a permutation so that the new series converges to any given value, or diverges.
https://en.wikipedia.org/wiki/Riemann_series_theorem
So any number you choose can be the limit of any conditionally convergent series, if it is permuted correctly.
A: To get a series that converges to $A$, pick any convergent series, say $$\sum_{k=1}^{\infty}b_k = B$$
and define $$a_k = \frac AB b_k$$
Then $$\sum_{k=1}^{\infty}a_k = A$$
A: If you want a series that sums to $a$, let $a_0=a$ and $a_n = 0$ for all other $n$.
If you want a series of nonzero terms that sums to $a$, let $a_n = \frac{a}{2^n}$ for $n\geq 1$.  Since 
$$
\sum_{n=1}^\infty \frac{1}{2^n} = 1
$$
we must have 
$$
\sum_{n=1}^\infty \frac{a}{2^n} = a
$$
Suppose a series $\sum b_n$ converges to $b$.  Let $a_n = \frac{a}{b} b_n$.  Then $\sum a_n$ converges to $a$:
$$
\sum a_n = \sum \frac{a}{b} b_n = \frac{a}{b} \sum b_n = \frac{a}{b}\,b = a
$$
Since there are infinitely many convergent series, there are infinitely many convergent series converging to $a$.
