Is the "inside term" $\frac{1}{n}- \ln \frac{n+1}{n}?$
If that is the case, there is a clear geometric interpretation. Draw the curve $y =1/x$. Draw also the following rectangles. They all have width $1$. The leftmost one has lower left corner at $(1,0)$, height $1$. The next one has lower left corner at $(2,0)$, height $1/2$. The next has lower left corner at $(3,0)$, height $1/3$. And so on.
When a partial sum of the harmonic series is approximated by a suitable definite integral of $1/x$, the "inside terms" represent the local approximation error, the error obtained by approximating the area of the rectangle by the area under $y=1/x$.
Added: In a not very interesting sense, every (convergent) series can be viewed as a telescoping sum. Consider the series $\sum_{n=1}^\infty a_n$. Let $s_n=\sum_{k=1}^n a_k$, and for convenience put $s_0=0$. Then $\sum_{n=1}^\infty a_n=(s_1-s_0)+(s_2-s_1)+(s_3-s_2) +\cdots.$
In particular, any sequence can be analyzed as coming from a telescoping series. If the sequence is to be reproduced term by term, then the obvious telescoping series representation is unavoidable. But there are many sophisticated ways of accelerating convergence that do not reproduce the sequence term by term, and for which a "telescoping series" description is not useful.