Max no. of keys of B-Tree of height h

Question

I am reading about B-Tree through CLRS. One of the exercise problem is to express maximum number of keys in a BTree of height $h$ and minimum degree $t$.

Properties of BTree of $t$:

• $t\ge 2$
• Min. no of keys in a node is $t-1$ and max. number of keys in a node is $2t-1$
• Max number of children a node can have is $2t$

To solve it the formula I came up with is $$\sum_0^h (2t)^h(2t-1) $$

Does it have any convergence?

Answer 1

Found it (using geometric serie for the sum)

$$\sum_{i=0}^h (2t)^i(2t-1) = (2t-1) \sum_{i=0}^h (2t)^i = (2t-1) \frac{(2t^{h+1}-1)}{(2t-1)} = (2t)^{h+1}-1 $$