Real Analysis Answers

Find the minimum amount of tin sheet in square inches that can be made into a closed cylinder having a volume of 108 cubic inches

Test the following series for convergence

∑

∞

n=1 [✓(n^4 +9) -✓(n^4 -9)]

Find lim x tends to 0 1-cos²x /x²sinx²

Test the following series for convergence 

∑
n=1
∞ [✓n^4+9 -✓n^4-9]

Prove that the sequence {an/n} is convergent where { an} is a bounded sequence

Show that x^4 + x^3 − 4x + 1 = 0 has a root in (0,1)

State Bonnet’s mean value theorem for integrals. Apply it to show that:

|₃∫⁵ cosxdx/x|≤ 2/3

Test the series:

_n=1 ∑∞ (-1)^n-1 [Sin(nx)]/n√n

for absolute and conditional convergence.

Give an example of a series  ∑a_n such that  ∑a_n is not convergent but the sequence (a_n) converges to 0.