I have got a positive series convergent series $w_j$, which is convergent. I can calculate a finite amount of terms $\sum_{j=0}^{j^*}w_j$ and would be interested in the estimation of the sum of the remaining terms $\sum_{j=j^*+1}^{\infty}w_j$.
I thought it would be nice doing so by using the property $s_k+\int_{k}^\infty f(x)dx\leq s\leq s_k+\int_{k+1}^\infty f(x)dx$. The series is not possible to integrate or differentiate, so I would replace it by a (extrapolation) polynomial $f(j)$.
I would like to interpolate the data points with a function that satisfies the properties:
- the function is integratable
- converges to 0 in infinity
- is positive and eventually $\forall j \geq j_0:\, f(j)\geq w_j$
I tried to create a Newton interpolation polynomial, which unfortunately diverges as seen in the picture:
Newton interpolation polynomial
I am not really interested in the interpolated values, but in the values for $\forall j \geq j_0:\, f(j)\geq w_j$
Thank you very much for any suggestions!