I have the series: $$\sum_{i=1}^{i=10^N} \log_5 i$$ I'm trying to figure out how to get the closed-form solution to this problem. I entered it into WolframAlpha and got that it equals: $ (\log(\mathrm{Pochhammer}(1, 10^N)))/(\log(5))$
I have a few things I'm not understanding about this problem,
- What procedure do you use to find a closed-form solution here?
- What does this solution even mean? (I've never seen this pochhammer symbol)
- If possible, could someone tell me if there's any general method for solving these types of problems? I have a lot of them and have been messing around with some algebra, but don't really even understand how to start them. It seems like I'm just having to google until I find some identity that looks similar enough in form. This is for a programming course, but I have no background in algorithms so I'm struggling a bit.
Use $\log_5(x)+\log_5(y)=\log_5(xy)$ to reduce the sum to $$\log_5(10^N!)$$
Here's the definition for Pochhammer symbol ( in this case rising factorial has been used ). $$x^{(n)}:=x(x+1)\cdots(x+n-1)$$ which implies $1^{(n)}=n!$ so our sum is $$\log_5(1^{(10^5)})$$ Wolfram alpha has a built-in function called Pochhammer to evaluate the rising factorial. $$\text{Pochhammer}[x,n]=x^{(n)}$$ So Wolfram alpha interprets our result as $$\log_5(\text{Pochhammer}[1,10^5])$$ But WA prefers base $e$ rather than any arbitrary base so using the identity $\log_ab={\log_c b\over\log_c a}$ for positive numbers $a,b,c;$ WA outputs the final result as $$ {\text{Log}[\text{Pochhammer}[1,10^5]]\over\text{Log[5]}} $$ where Log is the WA function function that calculates logarithm w.r.t $e$ unless we specify otherwise.