As simplification of TPID 12, we ask the much simpler question, whether
the sequence of interpolating polynomials for the points \( (0,0), (1,1), (2,2^{1/2}),\dots,(n,n^{1/n}) \) converges towards the function \( x^{1/x} \).
More precise:
Is \( \lim_{n\to\infty} f_n(x)=x^{1/x} \) for each \( x>0 \), where
\( f_N(x)=\sum_{n=0}^N \left(x\\n\right) \sum_{m=0}^n \left(n\\m\right) (-1)^{n-m} m^{1/m} \)?
a) Is that still true if we omit a certain number of points from the beginning of the sequence. For example omitting (0,0) and (1,1).
the sequence of interpolating polynomials for the points \( (0,0), (1,1), (2,2^{1/2}),\dots,(n,n^{1/n}) \) converges towards the function \( x^{1/x} \).
More precise:
Is \( \lim_{n\to\infty} f_n(x)=x^{1/x} \) for each \( x>0 \), where
\( f_N(x)=\sum_{n=0}^N \left(x\\n\right) \sum_{m=0}^n \left(n\\m\right) (-1)^{n-m} m^{1/m} \)?
a) Is that still true if we omit a certain number of points from the beginning of the sequence. For example omitting (0,0) and (1,1).