fn : [0, 1] → R, f„(x)= dt 1+t Show that for every x E [0, 1], this equality takes place: x2 In(1+ æ) = lim ( x 2 ..+(-1)"-1 3 nd then show that the sequence (an)n>1, is divergent, if: 1 1 1+ 2 1 an = + +...+ - 3 n
fn : [0, 1] → R, f„(x)= dt 1+t Show that for every x E [0, 1], this equality takes place: x2 In(1+ æ) = lim ( x 2 ..+(-1)"-1 3 nd then show that the sequence (an)n>1, is divergent, if: 1 1 1+ 2 1 an = + +...+ - 3 n
Chapter9: Sequences, Probability And Counting Theory
Section9.1: Sequences And Their Notations
Problem 71SE: Prove the conjecture made in the preceding exercise.
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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 $x \in[0,1]$, 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:
$$
\mathrm{a}_{\mathrm{n}}=1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n}
$$
Please write the complete solution and kindley take a look to the attached image.
Thanks in advance.
![I.Let n E N and the functions:
fn : [0, 1] → R,
fn (x) = Jo E
S dt
Show that for every x E [0, 1], this equality takes place:
In (1 + ӕ) -
x2
+
..+(-1)"--")
lim
n
and then show that the sequence (an)n>1, is divergent, if:
= 1+
+
3
+
an
...](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F55b0ccf5-d9c3-4a4a-afee-4882a9d37ea0%2Fb7ac59aa-4ca9-4866-837f-e11da1b3fe5e%2Fe55q3sn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:I.Let n E N and the functions:
fn : [0, 1] → R,
fn (x) = Jo E
S dt
Show that for every x E [0, 1], this equality takes place:
In (1 + ӕ) -
x2
+
..+(-1)"--")
lim
n
and then show that the sequence (an)n>1, is divergent, if:
= 1+
+
3
+
an
...
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