Q.1 - Calculus

Question

 true /false : determine whether ah of the statement that follow is true or false .if a statement is true , explain why. if a statement is false ,provide a counter example .

a) true or false : $\sum _{k=1}^{\infty }{\left(-1\right)}^{k+1} a_k$ is an alternating series

b) true or false : if $a_k>0 for k\in {\mathrm{\mathbb{Z}}}^{+}$ an the alternating series $\sum _{k=1}^{\infty }{\left(-1\right)}^k{a}_k$ converges, then the series $\sum _{k=1}^{\infty }{\left(-1\right)}^{k+1}{a}_k$ converges

c) true or false : if $a_k>0 for k\in {\mathrm{\mathbb{Z}}}^{+}$ an the series $\sum _{k=1}^{\infty }{a}_k$converges, then the series also converges absolutely .

d) true or false : if a function f satisfies the hypothesis of the integral test, then the series $\sum _{k=1}^{\infty }{\left(-1\right)}^{k+1}f\left(k\right)$ converges

e) true or false : if a series $\sum _{k=1}^{\infty }{a}_k$ converges conditionally , then the series $\sum _{k=1}^{\infty }\left|a_k\right|$ diverges.

f) true or false : if $\sum _{k=1}^{\infty }{a}_k$ is a series such that $\underset{k\to \infty }{\lim }\left|\frac{a_{k+1}}{a_k}\right|=1$ , then the series converges conditionally.

g) true or false : if $\sum _{k=1}^{\infty }{a}_k$is a series such that $\underset{k\to \infty }{\lim }\sqrt[k]{\left|a_k\right|}<1$ , then the series converges absolutely

h) true or false : if we rearrange infinitely many terms of the alternating harmonic series , we can change the value of its sum

Step-by-Step Solution

Answer

a)false

b)true

c)true

d)false

e)true

f)false

g)true

h)true

a) stp 1

consider the statement $\sum _{k=1}^{\infty }{\left(-1\right)}^{k+1}{a}_k$ is an alternating series

to determine whether the given statement is true or false

the series is alternating it an be written as 

$b_k={\left(-1\right)}^{k+1}{a}_{k } with a_k>0$.

the series $\sum _{k=1}^{\infty }{\left(-1\right)}^{k+1}{a}_k$ is alternating only if $a_k>0$.

if it is positive then the series $\sum _{k=1}^{\infty }{\left(-1\right)}^{k+1}{a}_k$ is alternative

then the given the statement is false

b) stp 1

consider the statement ,if $a_k>0 for k\in {\mathrm{\mathbb{Z}}}^{+}$ an the alternating series $\sum _{k=1}^{\infty }{\left(-1\right)}^k{a}_k$ converges,then the series $\sum _{k=1}^{\infty }{\left(-1\right)}^{k+1}{a}_k$ converges

to determine whether the given statement is true or false 

let $\left\{a_k\right\}$ be the sequence of positive numbers

if $a_{k+1}<a_{k } for vry k\ge 0 an \underset{k\to \infty }{\lim }{a}_k=0$

then , the series both$\sum _{k=1}^{\infty }{\left(-1\right)}^{k+1}{a}_k and \sum _{k=1}^{\infty }{(-1)}^k a_k$ convergent

b) step 2

if the series $\sum _{k=1}^{\infty }{\left(-1\right)}^k{a}_k$ is convergent then the series $\sum _{k=1}^{\infty }{\left(-1\right)}^{k+1}{a}_k$ is also convergent

hence the statement is true 

c) step 1

consider the statement , if $a_k>0 for k\in {\mathrm{\mathbb{Z}}}^{+}$an the series $\sum _{k=1}^{\infty }{a}_k$converges, then the series converges absolutely

to determine whether the given statement is true or false 

the series converges absolutely if $\sum _{k=1}^{\infty }\left|a_k\right|$ converges

c) step 2

if $a_k> o for k\in {\mathrm{\mathbb{Z}}}^{+}$ then the term of series $\sum _{k=1}^{\infty }{a}_k$ is positive

then $\left|a_k\right|=a_k$

hence $\sum _{k=1}^{\infty }\left|a_k\right|=\sum _{k=1}^{\infty }{a}_k$

since the series $\sum _{k=1}^{\infty }{a}_k$ is convergent then the series $\sum _{k=1}^{\infty }\left|a_k\right|$ is also convergent

if $a_k>0 for k\in {\mathrm{\mathbb{Z}}}^{+}$an the series $\sum _{k=1}^{\infty }{a}_k$ converges then the series converges absolutely

hence the statement is true 

d) step 1

if a function f satisfies the hypothesis of the integral test, then the series $\sum _{k=1}^{\infty }{\left(-1\right)}^{k+1}f\left(k\right)$ converges.

to determine whether given statement is true or false .

the series $\sum _{k=1}^{\infty }{\left(-1\right)}^{k+1}f\left(k\right)$ is a alternating series as f(k)>0 because f satisfies the hypothesis of the integral test 

d) step 2

the integral test only for positive terms but the series $\sum _{k=1}^{\infty }{\left(-1\right)}^{k+1}f\left(k\right)$ has a negative terms.

hence the integral part cannot be applied to series $\sum _{k=1}^{\infty }{\left(-1\right)}^{k+1}f\left(x\right)$

hence the statmnt is fals

)

if a sris $\sum _{k=1}^{\infty }{a}_k$onvrgs onitionally , thn th sris $\sum _{k=1}^{\infty }\left|a_k\right|$ is ivrgnt

to trmin whthr givn statmnt is tru or fals 

th sris$\sum _{k=1}^{\infty }{a}_k$ onvrgs onitionally .

by th finition , th sris $\sum _{k=1}^{\infty }{a}_k$ is onvrgnt

thn th sris $\sum _{k=1}^{\infty }\left|a_k\right|$ is ivrgnt baus $\sum _{k=1}^{\infty }{a}_k$ onvrgs onitionally.

hn th statmnt is tru 

f)

if $\underset{k=1}{\overset{\infty }{\sum a_k}}$ is a sris suh that $\underset{k\to \infty }{\lim }\left|\frac{a_{k+1}}{a_k}\right|=1$, thn th sris onvrgs onitionally.

to trmin whthr givn statmnt is tru or fals 

if $\underset{k\to \infty }{\lim }\left|\frac{a_{k+1}}{a_k}\right|=1$thn thratio tst bom inonlusiv

if $\underset{k\to \infty }{\lim }\left|\frac{a_{k+1}}{a_k}\right|= 1$ thn w annot b trmin th gonvrgn of th sris $\sum _{k=1}^{\infty }{a}_k$

hn th givn statmnt is fals.

g)

if $\sum _{k=1}^{\infty }{a}_k$ is a sris suh that $\underset{k\to \infty }{\lim }\left|\sqrt[k]{a_k}\right|<1$, thn th sris onvrgs onitionally .

to trmin whthr givn statmnt is tru or fals 

if $\underset{k\to \infty }{\lim }\left(\sqrt[k]{a_k}\right)<1$ thn th sris $\sum _{k=1}^{\infty }{a}_k$ is onvrgnt  by root tst

if $\underset{k\to \infty }{\lim }\left(\sqrt[k]{a_k}\right)<1$ thn th sris $\sum _{k=1}^{\infty }{a}_k$ onitionally onvrgnt by th root tst

hn givn statmnt is tru

h)

if w rarrang infinitly many trms of th altrnating harmoni sris ,w an hang th valu of its sum

to trmin whthr th givn statmnt is tru or fals 

onsir th sris for $\sum _{n=1}^{\infty }\frac{{(-1)}^{n+1}}{n}$

th valu of $\sum _{n=1}^{\infty }\frac{{(-1)}^{n+1}}{n}$is

$In\left(2\right)=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}$+...

th trms ar rarrang 

$$\begin{gathered} \sum _{n=1}^{\infty }\frac{{\left(-1\right)}^{n+1}}{n}=\left(1-\frac{1}{2}\right)-\frac{1}{4}+\left(\frac{1}{3}-\frac{1}{6}\right)-\frac{1}{8}+\left(\frac{1}{5}-\frac{1}{10}\right)+.... \\ =\frac{1}{2}-\frac{1}{4}+\frac{1}{6}-\frac{1}{8}+... \\ =\frac{1}{2}\left(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+...\right) \\ =\frac{1}{2}In 2 \end{gathered}$$

thus th rarrangmnt of infinitly many trms of th altrnating harmoni sris an hang th valu of sum .

hn th statmnt is tru.