I am new to math and will appreciate any help. Thank you!
$\begingroup$ I am also new to Mathematics on Stack Exchange, so if you see that I can improve my question, I would appreciate if you left a comment $\endgroup$
Commented Oct 29, 2017 at 19:24$\begingroup$ Demidovich. I've used it in 1977 to prepare Calculus I and after I continued to use it to give private classes to a lot of people, including the girl who became my wife. $\endgroup$
Commented Oct 29, 2017 at 20:58 $\begingroup$ Writing the first in binary will result in something interesting. $\endgroup$ Commented Oct 29, 2017 at 21:00$\begingroup$ @NikitaHismatov Once I asked a professor of mine about this. He replied: "Do you think we have a panacea to calculate limits?!" - As it often happens in mathematics, we have methods for some cases but you can always find harder limits for which the previously developed methods do not work. I once thought about the existence of the following limit and asked here, people were able to compute the limit but I'm not sure I really understand what they did. $\endgroup$
Commented Oct 30, 2017 at 6:34Hint:
All you have to know is the sum of the $n$ first terms of a geometric series, which is a formula from high school: $$1+q+q^2+\dots +q^n=\frac>\qquad (q\ne 1),$$from which we can deduce: $$q^r+q^+\dots +q^n=q^r(1+q+\dots+q^)=q^r\frac>=\frac>.$$ If $\lvert q\rvert<1$, these sums have limits $\;\dfrac1\;$ and $\;\dfrac\;$ respectively.
answered Oct 29, 2017 at 19:31 177k 10 10 gold badges 74 74 silver badges 177 177 bronze badges $\begingroup$What should I do when I need a limit of infinite sum? (are there any rules of thumb?)
In general - no there aren't such rules. There are specific types of series for which it is known how to compute the limit (like the geometric series). There are way more recepies for figuring out whether a series converges or not (finding the limit is much harder). But also here the cookbook is limited. As others pointed out already, your two series are geometric series, who's limit can be computed easily.
answered Oct 29, 2017 at 19:33 10.7k 14 14 silver badges 28 28 bronze badges $\begingroup$$$\color\frac 1 2 + \frac 1 4 + \frac 1 + \ldots + \frac >$$ Is an example of $$1+x+x^2+x^3+\ldots+x^n$$ Where the ratio between a number $a_n$ and the previous $a_$ is constant and is $x$. This is the sum of a geometric progression and it's quite easy to see that its value is $$\frac>$$ When $n\to\infty$ the sum converges only if $|x|\to 0$ and the sum is $$\sum_^<\infty>x^n=\frac$$ In your first example $x=\frac12$ and index $n$ starts from $n=1$ so the sum is $$\sum_^<\infty>\left(\frac12\right)^n=\frac-1=\color$$ The second one is $$\lim_