1
$\begingroup$

So I'm basically trying to write an explicit formula for this sum here:

Originally, I was asked to use the definition of an integral to evaluate: $$\int_{0}^{1} 2^x dx$$ so I rewrote this as a reimann sum to get:

$$\lim_{n\to\infty} {\frac {1}{n}}\sum_{i=1}^{n} 2^{\frac {i}{n}}$$

How do you evaluate that summation though? I'm completely lost on that. I tried to write it as a geometric series but that's not possible. It's not a p series either. How am I supposed to sum that?

Here is the way the professor did it but no work is shown for the summation at all... enter image description here

$\endgroup$
  • $\begingroup$ Do you know the indefinite integral of $2^x$? Note that $2^x = e^{x \ln 2}$, so maybe definition of integral could mean a use of Fundamental theorem of calculus? $\endgroup$ – астон вілла олоф мэллбэрг Dec 5 '16 at 5:49
  • $\begingroup$ No it explicitly asked us to use reimann sums. It's even in the solutions. But no explanation was given as to how that sum was obtained. $\endgroup$ – Future Math person Dec 5 '16 at 5:51
  • 1
    $\begingroup$ Then it's best you post the intermediate result, or the equality you don't understand in the answer. Surely there must be steps in the evaluation, right? $\endgroup$ – астон вілла олоф мэллбэрг Dec 5 '16 at 5:52
  • 1
    $\begingroup$ Fine, so the summation is a standard geometric series. Do you notice that? So the formula for geometric summation can be used. Do you need me to mention it explicitly? If so, then $\sum_{i=1}^n a^i = \frac{a^{n+1}-1}{a-1} - 1$. Put in $a=2^{\frac{1}{n}}$ and simplify. $\endgroup$ – астон вілла олоф мэллбэрг Dec 5 '16 at 5:55
  • 1
    $\begingroup$ It is a geometric series, right You are summing the powers of $2^{\frac 1n}$ from $i=1$ to $n$. It looks like a sequence where the ratio between consecutive terms is constant, right? Now the formula comes into play. $\endgroup$ – астон вілла олоф мэллбэрг Dec 5 '16 at 5:58

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.