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!