Real Analysis Answers

Using weiestrass M-test, show that the following series converges uniformly.

∞

∑n^3X^n,X belongs to[-1/3,1/3]

n=1

Using the definition, show' that the sequence [1/√n]neN is Cauchy.

Use the substitution theorem evaluate,

Integral 0 to 2 t^2[1+t^3]^-1/2dt

Apply second substitution theorem evaluate

Integral 1 to 4 dt/(|t+4|√t)

If f and g are continuous functions on [a,b] with integral from a to x f ≥ integral from a to x g for every x ∈ [a, b], must it be true that f(x) ≥ g(x) on [a, b]?

Apply second substitution theorem evaluate

i) integral 1 to 9 (√ t)/(2+√t)

Find the relative extrema for

i) 𝑓( 𝑥 )= 𝑥^3 − 3𝑥 + 5

𝑖𝑖) f(x)=𝑥^4 + 2𝑥^2 − 4

Determine whether the following functions are differentiable

i) 𝑓 (𝑥 )= |𝑥|;

ii) 𝑔(𝑥) = |𝑥| +| 𝑥 + 1 |

iii) h(x) = x^1/3

If I: = [0,4], calculate the norms of the following partitions:

a) P1: = (0,1,2,4)

b) P2: = (0,2,3,4)

c) P3: = (0,1,1.5,2,3.4,4)

d) P4: = (0,.5,2.5,3.5,4)

Test the following series for convergence

Σ from n=1to ♾️ for [√n⁴+9 - √n⁴-9]