Exercises: Series – "Math for Non-Geeks"

Math for Non-Geeks: Template:Aufgabe

Math for Non-Geeks: Template:Aufgabe

Math for Non-Geeks: Template:Aufgabe

Analogously, for the generalizes harmonic series ${\displaystyle \sum _{k=1}^{\infty }{\frac {1}{k^{\alpha }}}}$ with ${\displaystyle \alpha >1}$ we can show:

• ${\displaystyle \sum _{k=1}^{\infty }{\frac {1}{(2k-1)^{\alpha }}}=(1-{\frac {1}{2^{\alpha }}})\sum _{k=1}^{\infty }{\frac {1}{k^{\alpha }}}}$
• ${\displaystyle \sum _{k=1}^{\infty }{\frac {(-1)^{k+1}}{k^{\alpha }}}=(1-{\frac {1}{2^{\alpha -1}}})\sum _{k=1}^{\infty }{\frac {1}{k^{\alpha }}}}$

Math for Non-Geeks: Template:Aufgabe

Math for Non-Geeks: Template:Aufgabe

Math for Non-Geeks: Template:Aufgabe

Math for Non-Geeks: Template:Aufgabe