Calculating the number of terms in sequences

[Cheung]

How does one calculate the number of terms in the sequence

\sum\limits_{a=2}^k \sum\limits_{b=a}^k of 1/(a*b).

 

[Mark44]

How does one calculate the number of terms in the sequence

\sum\limits_{a=2}^k \sum\limits_{b=a}^k of 1/(a*b).

Is this what you're asking about?

$$ \sum_{a = 2}^k \sum_{b = a}^k \frac 1 {ab}$$

In any case, this is not a sequence, it's a sum (a double summation). To find how many terms, start by expanding the inner sum, and than expand the outer sum.

 

[HallsofIvy]

The "ath" term has k- a+ 1= k+1- a terms so there are $$\sum_{a= 2}^k (k+ 1- a)$$ We can write that as $$\sum_{a= 2}^k (k+1)- \sum_{a= 2}^k a$$. Of course, $$\sum{a= 2}^k (k+1)= (k-1)(k+1)$$. What is $$\sum_{a=2}^k a$$?