CARNEGIE MELLON UNIVERSITY
DEPARTMENT OF MATHEMATICAL SCIENCES
Integral Test Approximation Methods
Dr. Timothy Flaherty

For $p$-series, we may use quadratic approximations over the interval $[n+1,n+2]$, one quadratic interpolating the interval $[n,n+2]$ (an overestimate on $[n+1,n+2]$), and one for the interval $[n+1,n+3]$, (an underestimate on $[n+1,n+2]$). This yields:

Method 5: $s \approx A_n$, where $A_n=s_n+I_{n+1}+(1/24)a_n+(1/2)a_{n+1}-(1/24)a_{n+2}$

with $\vert E_n\vert\leq (1/12) f^{\prime \prime}(n)$.

As a comparison, for the p-series with $p=1.01$, and with error $\epsilon \leq 0.001$, we require only $6$ terms to approximate the sum using method 5; and just $55$ terms when for an error less than $10^{-6}$.

Hot off the press! Methods 6 and 7 - stayed tuned, we can approximate with error controlled by either the third or forth derivative! I'm on to something, but have a lot of algebra to do before establishing the general result: an approximation by a "partial sum" and an improper integral, with error less than a constant times the $m^{\rm th}$ derivative.