In this paper,via the beta function we evaluate some series of the type∑^(∞)_(k=0)(ak+b)x^(k)/(^(mk)_(nk)).For example,we prove that^(∞)∑_(k=0)(49k+1)8^(k)/3^(k)(^(3k)_(k))=81+16√3πand^(∞)∑_(k=0)(10k-1)/(^(4k)...In this paper,via the beta function we evaluate some series of the type∑^(∞)_(k=0)(ak+b)x^(k)/(^(mk)_(nk)).For example,we prove that^(∞)∑_(k=0)(49k+1)8^(k)/3^(k)(^(3k)_(k))=81+16√3πand^(∞)∑_(k=0)(10k-1)/(^(4k)_(2k))=4√3/27π.We also establish the following efficient formula for computing logn with 1<n≤85/4:^(∞)∑_(k=0)(2(n^(2)+6n+1)^(2)(n^(2)-10n+1)k+P(n))(n-1)^(4k)/(-n)^(k)(n+1)^(2k)(^(4k)_(2k))=6n(n+1)(n−1)^(3)log n−32n(n+1)^(2)(n^(2)−4n+1),where P(n):=n^(6)-58n^(5)+159n^(4)+52n^(3)+159n^(2)-58n+1.In addition,we.pose some con]ectures on series w hose summands involve (^(2k)_(k))/((^(3k)_(k))(^(6k)_(3k)))(k∈N).展开更多
基金Supported by the National Natural Science Foundation of China(grant no.12371004)。
文摘In this paper,via the beta function we evaluate some series of the type∑^(∞)_(k=0)(ak+b)x^(k)/(^(mk)_(nk)).For example,we prove that^(∞)∑_(k=0)(49k+1)8^(k)/3^(k)(^(3k)_(k))=81+16√3πand^(∞)∑_(k=0)(10k-1)/(^(4k)_(2k))=4√3/27π.We also establish the following efficient formula for computing logn with 1<n≤85/4:^(∞)∑_(k=0)(2(n^(2)+6n+1)^(2)(n^(2)-10n+1)k+P(n))(n-1)^(4k)/(-n)^(k)(n+1)^(2k)(^(4k)_(2k))=6n(n+1)(n−1)^(3)log n−32n(n+1)^(2)(n^(2)−4n+1),where P(n):=n^(6)-58n^(5)+159n^(4)+52n^(3)+159n^(2)-58n+1.In addition,we.pose some con]ectures on series w hose summands involve (^(2k)_(k))/((^(3k)_(k))(^(6k)_(3k)))(k∈N).
