<s>[INST] How do I express the sum $(1+k)+(1+k)^2+\ldots+(1+k)^N$ for $|k|\ll1$ as a series? Wolfram Alpha provides the following exact solution
$$ \sum_{i=1}^N (1+k)^i = \frac{(1+k)\,((1+k)^N-1)}{k}.$$
I wish to solve for $N$ of the order of several thousand and $|k|$ very small (c. $10^{-12}).$  When I do this on a computer in excel the software cannot handle it (because of truncation of significant figures) and the results are nonsense.
I thought to approximate the result using the first few terms of a series in increasing powers of $k$.  I can multiply out the first few terms and examine the patterns in the following pyramid...
$$(1+k)^1 =               k +1  $$
$$(1+k)^2 =        k^2  +2k +1  $$
$$(1+k)^3 = k^3  +3k^2  +3k +1  $$
$$(1+k)^4 =k^4+4 k^3+6 k^2+4 k+1$$
$$(1+k)^5 =k^5+5 k^4+10 k^3+10 k^2+5 k+1$$
$$(1+k)^6 =k^6+6 k^5+15 k^4+20 k^3+15 k^2+6 k+1$$
So for example, for $N=3$ we would obtain the sum
$$
S= k^3  +4k^2 +6k +1
$$
The results suggest a solution with a pattern of the form
$$
S = a + bk^1 +ck^2+dk^3...
$$
I can see that $a=N$.  The other coefficients increase monotonously and it might be possible to determine a formula for the coefficients from the pattern. Although the general pattern is not convergent, it is possible that for certain restricted ranges of $N$ and $k$ a convergent formula could be obtained.  If so then it is possible that a useful approximation of S can be obtained just by evaluating the first few terms in the series.
But is there a well-known general formula for the terms in this series or can one be derived algebraically from the original formula?

UPDATE
following on from the answer by User73985...
$$ S=\sum_{i=1}^N (1+k)^i = N + \sum_{j=2}^{N+1}\binom{N+1}{j}k^{j-1}$$
So
$$ S= N + \sum_{j=2}^{N+1}\frac{(N+1)!}{(N+1-j)!j!}k^{j-1}$$
then
$$ S= N 
+  \frac{(N+1)!}{(N-1)!2!} k^{1} 
+  \frac{(N+1)!}{(N-2)!3!} k^{2}
+  \frac{(N+1)!}{(N-3)!4!} k^{3} +...
$$
giving
$$ S= N 
+  \frac{(N+1)(N)}{2!} k^{1} 
+  \frac{(N+1)(N)(N-1)}{3!} k^{2}
+  \frac{(N+1)(N)(N-1)(N-2)}{4!} k^{3} +...
$$
thus
$$ S= N 
+  \frac{N^2+N}{2} k^{1} 
+  \frac{N^3-N}{6} k^{2}
+  \frac{N^4-2 N^3-N^2+2 N}{24} k^{3} +...
$$
For $N=1 to 10,000$ and $k= 2.40242 * 10^{-12}$ this formula can be truncated to
$$ S = N + \frac{N^2+N}{2} k^{1} 
$$
and then gives results very close to those expected. Because $k$ is so small relative to $n$ the terms in higher powers of $k$ can be ignored. Note that the coefficient of $k^1$ is consistent with that found by examination of the coefficients in the "pyramid" presented above.
 [/INST] for $k<1$ we can use
$$
(1+k)^N \approx 1+ kN + \frac{N(N-1)}{2}k^2 + \frac{N(N-1)(N-2)}{3!}k^3
$$
thus
$$
\begin{align}
\sum (1+k)^i &\approx&\frac{(k+1)(1+ kN + \frac{N(N-1)}{2}k^2 + \frac{N(N-1)(N-2)}{3!}k^3-1)}{k}\\
&=&(k+1)\left(1+\frac{N-1}{2}k+\frac{(N-1)(N-2)}{3!}k^2\right)
\end{align}
$$
</s>