Hello everyone, this is my first post on this forum with this question. I don't understand how the binomials are simplified down from the second step to the third step to ultimately get 1/15^15. Clarification is greatly appreciated.
The problem:
. . . . .\(\displaystyle \displaystyle \sum_{n=1}^{\infty}\, \dfrac{(n!)^{15}}{(15n)!}\)
The steps after applying the ratio test:
. . .\(\displaystyle \displaystyle \rho\, =\, \lim_{n \rightarrow \infty}\, \left| \, \dfrac{((n\, +\, 1)!)^{15}}{(15(n+1))!}\, \dfrac{(15n)!}{(n!)^{15}}\, \right| \)
. . . . .\(\displaystyle \displaystyle =\, \lim_{n \rightarrow \infty}\, \left| \, \dfrac{(n\, +\, 1)^{15}}{(15n\, +\, 1)\, \cdot\, ...\, \cdot\, (15n\, +\, 15)}\, \right| \)
. . . . .\(\displaystyle \displaystyle =\, \lim_{n \rightarrow \infty}\, \left| \, \dfrac{n^{15}\, +\, ...}{15^{15}\, n^{15}\, +\, ...} \right| \)
. . . . .\(\displaystyle \displaystyle =\, \dfrac{1}{15^{15}}\)
The problem:
. . . . .\(\displaystyle \displaystyle \sum_{n=1}^{\infty}\, \dfrac{(n!)^{15}}{(15n)!}\)
The steps after applying the ratio test:
. . .\(\displaystyle \displaystyle \rho\, =\, \lim_{n \rightarrow \infty}\, \left| \, \dfrac{((n\, +\, 1)!)^{15}}{(15(n+1))!}\, \dfrac{(15n)!}{(n!)^{15}}\, \right| \)
. . . . .\(\displaystyle \displaystyle =\, \lim_{n \rightarrow \infty}\, \left| \, \dfrac{(n\, +\, 1)^{15}}{(15n\, +\, 1)\, \cdot\, ...\, \cdot\, (15n\, +\, 15)}\, \right| \)
. . . . .\(\displaystyle \displaystyle =\, \lim_{n \rightarrow \infty}\, \left| \, \dfrac{n^{15}\, +\, ...}{15^{15}\, n^{15}\, +\, ...} \right| \)
. . . . .\(\displaystyle \displaystyle =\, \dfrac{1}{15^{15}}\)
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