MathDB

Problem 4-M

Part of 1997 VJIMC

Problems(2)

# of 0's in decimal expansion, summation converges

Source: VJIMC 1997 1.4-M

9/30/2021
Find all real numbers a>0a>0 for which the series ∑n=1∞af(n)n2\sum_{n=1}^\infty\frac{a^{f(n)}}{n^2}is convergent; f(n)f(n) denotes the number of 00's in the decimal expansion of ff.
limitsreal analysisnumber theoryDigits
infinite factorial sum with trig expression

Source: VJIMC 1997 2.4-M

10/9/2021
Prove that ∑n=1∞n2(7n)!=173∑k=12∑j=06ecos⁡(2πj/7)⋅cos⁡(2kπj7+sin⁡2πj7).\sum_{n=1}^\infty\frac{n^2}{(7n)!}=\frac1{7^3}\sum_{k=1}^2\sum_{j=0}^6e^{\cos(2\pi j/7)}\cdot\cos\left(\frac{2k\pi j}7+\sin\frac{2\pi j}7\right).
trigonometrySummationreal analysislimitsfactorial