Thread: Series induction (real analysis)

2 questions

#1) prove by induction that 1/k! is less than or equal to 1/(2^k-1))

I showed the base case and assumed it to be true for all k and want to show that it is true for k+1 or that 1/(k+1)! <= 1/2^k

From my assumption I can either multiply both sides by 1/2 and get that 1/2*k! <= 1/(2^k) which gets me the correct rhs or I can multiply both sides by 1/(k+1) to get the correct lhs side but can't get both at once, any suggestions?

#2) Let s_n = the sum from k=1 to n of 1/k! and t_n = the sum from k=1 to n of 1/(2^k-1)
Prove by induction that s_n<=t_n


No idea how to do this after showing my base and assuming true for n any help would be great

Thanks a lot!