Calculus: Improper integrals vs series What is the difference between improper integrals and the a series?
For example, if you solve a type one improper integral from 1 to infinity, the answer is different than if you solve the same function using a geometric series.
In your case, taking $1 \to \infty$: 

Series sum over integers $n = 1 \to \infty$: $\displaystyle \sum_{n=1}^\infty f(n)$. 

Integrals "sum" over all reals $x = 1 \to \infty$: $\int_{n=1}^\infty f(x) dx$.



Added: this might help in understanding the relationship between partial summation (series) and integration, from Wikipedia's entry on "Series": 


  Partial summation takes as input a sequence, $\{ a_n \}$, and gives as output another sequence, $\{ S_N \}$. It is thus a unary operation on sequences. Further, this function is linear, and thus is a linear operator on the vector space of sequences, denoted $\Sigma$. The inverse operator is the finite difference operator, $\Delta$. These behave as discrete analogs of integration and differentiation, only for series (functions of a natural number) instead of functions of a real variable. For example, the sequence $\{1, 1, 1, ...\}$ has series $\{1, 2, 3, 4, ...\}$ as its partial summation, which is analogous to the fact that $\int_0^x 1\,dt = x.$
More clearly, for series  we use $\sum$ which are discrete values summed over integers and for integrals we use $\int_a^b$ which are continuous interval