The number of increasing sequences of elements of (\{1, ..., n\}) of length (k) and ending in (m) is (\binom{m-1}{k-1}) (pick the (k-1) elements less than (m) from (\{1, ..., m-1\}) and there's a unique way to order them). There are (m) ways in which the following roll won't improve on the last one. The probability of this scenario is then (\frac{1}{n^{k+1}}\binom{m-1}{k-1}m=\frac{k}{n^{k+1}}\binom{m}{k}).
The desired probability is then (\sum_{m=1}^n m\sum_{k \text{ odd}} \frac{1}{n^{k+1}}\binom{m-1}{k-1}).
To compute the inner sum, we write it as (\frac{1}{n^2}\sum_{k \text{ odd}} \frac{1}{n^{k-1}}\binom{m-1}{k-1}=\frac{1}{n^2}\left[\large(1+\frac{1}{n}\right)^{m-1}-\large(1-\frac{1}{n}\right)^{m-1}\right]).
The probability then becomes
(\frac{1}{n^2}\sum_{m=1}^n\left[m\large(1+\frac{1}{n}\right)^{m-1}-m\large(1-\frac{1}{n}\right)^{m-1}\right]).
To compute the latter two sums, note that they are both of the form (\sum_{m=1}^n mx^{m-1}), which is the derivative of (\sum_{m=1}^n x^m = \frac{x^{n+1}-x}{x-1}).