Maximizing Limits and Integrals: Solving Three Challenging Problems

[MathematicalPhysicist]

i have three questions:
1) find the limit of $$b_n=\frac{1}{\sqrt n^2}+\frac{1}{\sqrt(n^2-1)}+...+\frac{1}{\sqrt(n^2-(n-1)^2)}$$
2) if a is any number greater than -1, evaluate $$\lim_{n\rightarrow\infty} \frac{1^a+2^a+...+n^a}{n^{a+1}}$$
3) prove that $$\int_{0}^{x}[\int_{0}^{u}f(t)dt]du=\int_{0}^{x}f(u)(x-u)du$$

for the first i got: half pi, and for the second question i got 1/(a+1) is this correct?

for the third question, here's what i did:
$$\int_{0}^{x}u'[\int_{0}^{u}f(t)dt]du=[\int_{0}^{u}f(t)dtu]_{0}^{x}-\int_{0}^{x}uf(u)du$$ now my question is can i use here a change of dummy variable here for the first integral, from f(t)dt to f(u)du and to get the equality?

 

[MathematicalPhysicist]

no one has got anything to say?

 

[quasar987]

How did you get the first one?

 

[MathematicalPhysicist]

i used riemann sums here, we have the sum:
$$\sum_{k=1}^{n-1}\frac{1}{n}\frac{1}{\sqrt{1-(\frac{k}{n})^2}}+\frac{1}{n}$$
is this correct?

 

[nocturnal]

i have three questions:
1) find the limit of $$b_n=\frac{1}{\sqrt n^2}+\frac{1}{\sqrt(n^2-1)}+...+\frac{1}{\sqrt(n^2-(n-1)^2)}$$
2) if a is any number greater than -1, evaluate $$\lim_{n\rightarrow\infty} \frac{1^a+2^a+...+n^a}{n^{a+1}}$$
3) prove that $$\int_{0}^{x}[\int_{0}^{u}f(t)dt]du=\int_{0}^{x}f(u)(x-u)du$$

for the first i got: half pi, and for the second question i got 1/(a+1) is this correct?

for the third question, here's what i did:
$$\int_{0}^{x}u'[\int_{0}^{u}f(t)dt]du=[\int_{0}^{u}f(t)dtu]_{0}^{x}-\int_{0}^{x}uf(u)du$$ now my question is can i use here a change of dummy variable here for the first integral, from f(t)dt to f(u)du and to get the equality?

Your first two answers look good.

For the third, I can't make sense of what you've done. What is u'?
Heres a hint:
Define the functions F and G as

$$F(x) = \int_{0}^{x} \left( \int_{0}^{u} f(t)dt \right) du$$

$$G(x) = \int_{0}^{x}f(u)(x-u)du$$

Find the derivatives of F and G with respect to x. Deduce from this that there is a constant C such that F = G + C.

 

[MathematicalPhysicist]

u' is the derivative of u wrt u.
i.e du/du=1.

 

[nocturnal]

Oh I see, you used the product rule (integration by parts). That'll work too!

Sure, you can always substitute the dummy variable so long as its different from the one used for the limit of integration, so in this case you'd have to first evaluate the expression,

$$\left[ \int_{0}^{u}uf(t)dt \right]_{0}^{x} = \int_{0}^{x} x f(t)dt$$

and then make the substitution.