I.Let n E N and the functions: fn : [0, 1] → R, fn(x) = So dt 1+t a. Show that for every x E [0, 1], f,(X) < X b. Show that for every x E 0, 1] and, 1 0 < f (x) < n +1 Calculate: lim fn(x). n 00 c. Show that for every x E 0, 1| and everyn e N*, x' fn(x) + fn-1(x) n d. Show that for every x E [0, 1], this equality takes place: x3 + 3 x2 In(1+ x) lim +(-1)"-1 .. n and then show that the sequence (an)n>1, is divergent, if: 1 1+ 1 an 3 n
Let $n \in \mathbf{N}$ and the functions:
$f_{n}:[0,1] \rightarrow \mathbf{R},\quad$ $\mathrm{f}_{\mathrm{n}}(\mathrm{x})=\int_{0}^{x} \frac{t^{n}}{1+t} d t$
a. Show that for every $x \in[0,1]$,
$$
f_{n}^{\prime}(\mathrm{X}) \leq \mathrm{X}
$$
b. Show that for every $x \in[0,1]$ and,
$$
0 \leq \mathrm{f}_{\mathrm{n}}(\mathrm{x}) \leq \frac{1}{n+1}
$$
Calculate:
$$
\lim _{n \rightarrow \infty} f_{n}(x) \text {. }
$$
c. Show that for every $\mathrm{x} \in[0,1]$ and every $\mathrm{n} \in \mathbf{N}^{*}$,
$$
f_{n}(x)+f_{n-1}(x)=\frac{x^{n}}{n} .
$$
d. Show that for every $x \in[0,1]$, this equality takes place:
$$
\ln (1+x)=\lim _{n \rightarrow \infty}\left(x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\ldots+(-1)^{n-1} \frac{x^{n}}{n}\right)
$$ and then show that the sequence $\left(a_{n}\right)_{n \geq 1}$, is divergent, if:
$$
\mathrm{a}_{\mathrm{n}}=1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n}
$$
This is one single question with 4 subpoints, please write a complete answer.
Please take a look to the attached picture for details.
Thanks in advance.
![I.Let n E N and the functions:
fn : [0, 1] → R,
fn (x) = Jo 1HE
a. Show that for every x E 0, 1],
f.(X) < X
b. Show that for every x E 0, 1| and,
1
0< fn (x) <
n +1
Calculate:
lim fn(x).
c. Show that for every x E 0,1| and every n e N*,
fn (x) + fn-1(x)
n
d. Show that for every x E [0, 1], this equality takes place:
x2
x3
In(1+ x) = lim
+(-1)"-1,
2
3
n
and then show that the sequence (an)n>1, is divergent, if:
1
1+
1
an
3
n](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F55b0ccf5-d9c3-4a4a-afee-4882a9d37ea0%2F8b1e82f4-7855-4d30-94bf-9919186f4b92%2F3lxlq0q_processed.jpeg&w=3840&q=75)
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