Proof?
The semantics is exactly the same, and compiler is excellent at figuring out this kind of things.

Edit: I tried to compile this version and the same with switches instead of if-else. On my compiler (gcc7.3.0) both versions compile to the exact same instruction sequence.

Any addition might overflow/underflow, including operator +.

Why shouldn't the last 1 be a cubic number?

Why not?

Your pow function might run enlessly or return incorrect results because your y parameter is a double.

Careful: whitespace does not imply only " ".

never returns for n<0

rand() doesn't generate real random numbers, and using it with modulo introduces bias.
I recommend using utilities in the header (with std::uniform_int_distribution) if you need quality random numbers.
std::random_device can provide real random numbers if your system has entropy.

For data, you can construct a vector (or std::array) with fixed size 255.

switch is faster than if-else

forgot about it )

look up std::bitset

look up std::bitset

hint:

1. either sum up the smallest values first for better precision, then you don't need temp
2. or stop after temp < sum * std::numeric_limits::epsilon.
   Any further term wont increase sum.