# 6.4 Working with taylor series  (Page 3/11)

 Page 3 / 11

## Deriving maclaurin series from known series

Find the Maclaurin series of each of the following functions by using one of the series listed in [link] .

1. $f\left(x\right)=\text{cos}\sqrt{x}$
2. $f\left(x\right)=\text{sinh}\phantom{\rule{0.1em}{0ex}}x$
1. Using the Maclaurin series for $\text{cos}\phantom{\rule{0.1em}{0ex}}x$ we find that the Maclaurin series for $\text{cos}\sqrt{x}$ is given by
$\begin{array}{cc}\hfill \sum _{n=0}^{\infty }\frac{{\left(-1\right)}^{n}{\left(\sqrt{x}\right)}^{2n}}{\left(2n\right)\text{!}}& =\sum _{n=0}^{\infty }\frac{{\left(-1\right)}^{n}{x}^{n}}{\left(2n\right)\text{!}}\hfill \\ & =1-\frac{x}{2\text{!}}+\frac{{x}^{2}}{4\text{!}}-\frac{{x}^{3}}{6\text{!}}+\frac{{x}^{4}}{8\text{!}}-\text{⋯}.\hfill \end{array}$

This series converges to $\text{cos}\sqrt{x}$ for all $x$ in the domain of $\text{cos}\sqrt{x};$ that is, for all $x\ge 0.$
2. To find the Maclaurin series for $\text{sinh}\phantom{\rule{0.1em}{0ex}}x,$ we use the fact that
$\text{sinh}\phantom{\rule{0.1em}{0ex}}x=\frac{{e}^{x}-{e}^{\text{−}x}}{2}.$

Using the Maclaurin series for ${e}^{x},$ we see that the $n\text{th}$ term in the Maclaurin series for $\text{sinh}\phantom{\rule{0.1em}{0ex}}x$ is given by
$\frac{{x}^{n}}{n\text{!}}-\frac{{\left(\text{−}x\right)}^{n}}{n\text{!}}.$

For $n$ even, this term is zero. For $n$ odd, this term is $\frac{2{x}^{n}}{n\text{!}}.$ Therefore, the Maclaurin series for $\text{sinh}\phantom{\rule{0.1em}{0ex}}x$ has only odd-order terms and is given by
$\sum _{n=0}^{\infty }\frac{{x}^{2n+1}}{\left(2n+1\right)\text{!}}=x+\frac{{x}^{3}}{3\text{!}}+\frac{{x}^{5}}{5\text{!}}+\text{⋯}.$

Find the Maclaurin series for $\text{sin}\left({x}^{2}\right).$

$\sum _{n=0}^{\infty }\frac{{\left(-1\right)}^{n}{x}^{4n+2}}{\left(2n+1\right)\text{!}}$

We also showed previously in this chapter how power series can be differentiated term by term to create a new power series. In [link] , we differentiate the binomial series for $\sqrt{1+x}$ term by term to find the binomial series for $\frac{1}{\sqrt{1+x}}.$ Note that we could construct the binomial series for $\frac{1}{\sqrt{1+x}}$ directly from the definition, but differentiating the binomial series for $\sqrt{1+x}$ is an easier calculation.

## Differentiating a series to find a new series

Use the binomial series for $\sqrt{1+x}$ to find the binomial series for $\frac{1}{\sqrt{1+x}}.$

The two functions are related by

$\frac{d}{dx}\sqrt{1+x}=\frac{1}{2\sqrt{1+x}},$

so the binomial series for $\frac{1}{\sqrt{1+x}}$ is given by

$\begin{array}{cc}\hfill \frac{1}{\sqrt{1+x}}& =2\frac{d}{dx}\sqrt{1+x}\hfill \\ & =1+\sum _{n=1}^{\infty }\frac{{\left(-1\right)}^{n}}{n\text{!}}\phantom{\rule{0.2em}{0ex}}\frac{1·3·5\text{⋯}\left(2n-1\right)}{{2}^{n}}{x}^{n}.\hfill \end{array}$

Find the binomial series for $f\left(x\right)=\frac{1}{{\left(1+x\right)}^{3\text{/}2}}$

$\sum _{n=1}^{\infty }\frac{{\left(-1\right)}^{n}}{n\text{!}}\phantom{\rule{0.2em}{0ex}}\frac{1·3·5\text{⋯}\left(2n-1\right)}{{2}^{n}}{x}^{n}$

In this example, we differentiated a known Taylor series to construct a Taylor series for another function. The ability to differentiate power series term by term makes them a powerful tool for solving differential equations. We now show how this is accomplished.

## Solving differential equations with power series

Consider the differential equation

${y}^{\prime }\left(x\right)=y.$

Recall that this is a first-order separable equation and its solution is $y=C{e}^{x}.$ This equation is easily solved using techniques discussed earlier in the text. For most differential equations, however, we do not yet have analytical tools to solve them. Power series are an extremely useful tool for solving many types of differential equations. In this technique, we look for a solution of the form $y=\sum _{n=0}^{\infty }{c}_{n}{x}^{n}$ and determine what the coefficients would need to be. In the next example, we consider an initial-value problem involving ${y}^{\prime }=y$ to illustrate the technique.

## Power series solution of a differential equation

Use power series to solve the initial-value problem

${y}^{\prime }=y,\phantom{\rule{0.5em}{0ex}}y\left(0\right)=3.$

Suppose that there exists a power series solution

$y\left(x\right)=\sum _{n=0}^{\infty }{c}_{n}{x}^{n}={c}_{0}+{c}_{1}x+{c}_{2}{x}^{2}+{c}_{3}{x}^{3}+{c}_{4}{x}^{4}+\text{⋯}.$

Differentiating this series term by term, we obtain

${y}^{\prime }={c}_{1}+2{c}_{2}x+3{c}_{3}{x}^{2}+4{c}_{4}{x}^{3}+\text{⋯}.$

If y satisfies the differential equation, then

${c}_{0}+{c}_{1}x+{c}_{2}{x}^{2}+{c}_{3}{x}^{3}+\text{⋯}={c}_{1}+2{c}_{2}x+3{c}_{3}{x}^{2}+4{c}_{3}{x}^{3}+\text{⋯}.$

Using [link] on the uniqueness of power series representations, we know that these series can only be equal if their coefficients are equal. Therefore,

$\begin{array}{c}{c}_{0}={c}_{1},\hfill \\ {c}_{1}=2{c}_{2},\hfill \\ {c}_{2}=3{c}_{3},\hfill \\ {c}_{3}=4{c}_{4},\hfill \\ \hfill \text{⋮}.\hfill \end{array}$

Using the initial condition $y\left(0\right)=3$ combined with the power series representation

$y\left(x\right)={c}_{0}+{c}_{1}x+{c}_{2}{x}^{2}+{c}_{3}{x}^{3}+\text{⋯},$

we find that ${c}_{0}=3.$ We are now ready to solve for the rest of the coefficients. Using the fact that ${c}_{0}=3,$ we have

$\begin{array}{}\\ \\ {c}_{1}={c}_{0}=3=\frac{3}{1\text{!}},\hfill \\ {c}_{2}=\frac{{c}_{1}}{2}=\frac{3}{2}=\frac{3}{2\text{!}},\hfill \\ {c}_{3}=\frac{{c}_{2}}{3}=\frac{3}{3·2}=\frac{3}{3\text{!}},\hfill \\ {c}_{4}=\frac{{c}_{3}}{4}=\frac{3}{4·3·2}=\frac{3}{4\text{!}}.\hfill \end{array}$

Therefore,

$\begin{array}{cc}\hfill y& =3\left[1+\frac{1}{1\text{!}}x+\frac{1}{2\text{!}}{x}^{2}+\frac{1}{3\text{!}}{x}^{3}\frac{1}{4\text{!}}{x}^{4}+\text{⋯}\right]\hfill \\ & =3\sum _{n=0}^{\infty }\frac{{x}^{n}}{n\text{!}}.\hfill \end{array}$

You might recognize

$\sum _{n=0}^{\infty }\frac{{x}^{n}}{n\text{!}}$

as the Taylor series for ${e}^{x}.$ Therefore, the solution is $y=3{e}^{x}.$

what is microbiology
microbiology is the study of micro organisms. this can be multicellular,unicellular & acellular
Janet
what is the difference between DNA/RNA
kanaaLka
DNA is a double stranded molecule while RNA is a single stranded molecule
Lovely
who is the inventor of microscope
Alberta
What are antibodies?
they are immune cells that are released by immune system to fight strangers like microbes
Ahmed
a blood protien produce in response to an counteracting a specific antigen
Pooja
Why salmonella typhi is harmful protozoan?
because make typhoid that is considered as a food born illness
Ahmed
Name any five modes of transmission.
Happy
what are the scientific names and common names of some microbes
Staphylococcus aureus - staph Streptococcus pyogenes - strep Botulism - Clostridium botulinum Rocky Mountain spotted fever - Rickettsia rickettsii Diphtheria - Corynebacterium diphtheriae Gonorrhea - Neisseria gonorrhoeae
Tammy
what are the limitations of the Koch's postulate
Naa
how a microorganism growth.
Is foetus a parasite to his or her mother? if yes and if no ho
y
Yusri
yes
Yashkin
yes
Zubair
yes it a parasite to the mother because it feeds on the mother for survival
Beatrice
yes
Redwan
no
Elasha
definition of a parasite: an organism that lives in or on an organism of ANOTHER species (its host) and benefits by deriving nutrients at the other's expense
Elasha
no because an organism can inky be a parasite if it causes harm to its HOST. And the fetus does not cause harm to its mother under normal conditions
aliyu
yes
Naa
what is biosensor in microbiology
What are biosensor
Raja
what's are biosensor
Raja
what is microbiology defination
microbiology is the study of small or manuit organisms which cannot be seen with our nacked eyes unless with the aid of the microscope
Brandina
is the study of living organisms which are not directly visible to a direct eye but can only be seen under a microscope
Ipa
ok
Ipa
microbiology is the study of living organisms of microscopic size it is also the study of micro organisms with their form structure reproduction psychology metabolism and classification
Priyanka
ok
Aminu
Microbiology are the study of microorganisms either microscopic or sub microscopic creaters mainly unicellulars, multicellulars and subcellulars. Such as protozoa,bacteria and viruses.
Rana
it is the biological study of viruses, fungi, protozoa, bacteria which in collective name are called micro organisms, unlike microscopic organisms being invisible, that requires a magnifications with the help of a microscope.
Mohamed
what are the importances of Microbiology?
don't know
SINGLE
we obtained the insulin from the bacteria. and some microorganisms are decomposer in ecosystems.
Zubair
nitrogen in the air is fixed into the soil by microbes example is nitrobactor
Matilda
yes
Zubair
how a weak immune region where microorganisms attack easily?
with examples differentiate gram positive from gram negative bacteria
Differentiate gram positive from gram negative
Mary
I have no idea
Zubair
hello
Kuonain
and example of gram negative is E. coli
Pooja
gram positive stain purple when subjected to gram stain whilst gram positive bacterial has thick wall composed of peptidoglycan
Matilda
ok
Zubair
what is a process of gene expression in eukaryotes ?
pls help us with the answer
Abdussalam
spontaneous generation means
hiii
Siddhi
Hello
eman
So, what's next 😂
eman
I have a problem with Micro
eman
I studying things and see question Some thing else
eman
eman
What i havta do
eman
firsfall when you studying understand things...
Siddhi
good afternoon dear Friends
yaya
Greeting to everyone in here.
Manka
There is a problem. I need a diagram of a virus with it functions.
Manka
structure of bacterial
Kuyiba
gghhhh
Zubair
group plz i need help in microbio
Kuonain
it really difficult fr me
Kuonain
kashur cha kah
suhail
sorry now I read in class 8th but I can help u
Zubair
Me too. It's very difficult for me
Angela
rod shape
Priyanka
helical
Priyanka
Hello
esike
spherical
Priyanka
peomorphic
Priyanka
I am ryt or not
Priyanka
hi
suhail
hello
DIPTI
Are we together pls what's the topic for the day?
esike
what is innate
Lizzy
innate means natural
Pooja
the genetic makeup of an individual
Matilda
pls i need the common names for the following parasites..ENTEROBIUS VERMICULARIS,NECATOR AMERICANUS,ASCARIES LUMBRICOIDES,TRICHURIS TRICHIURA,TRICHOMONAS VAGINALIS,GIARDIA LAMBLIA,ENTAMOEBA HISTOLYTICA,SCHISTOSOMA MANSONI,SCHISTOSOMAHEMATOBIUM,STROGYLOIDES STERCORALIS,AND TRAPANOSOMA BRUCI GAMBIENSE
timothy
ENTEROBIUS VERMICULARIS and STROGYLOIDES STERCORALIS have the same common name which is pin worm and thread worm
Chinedu
I didn't find the common names for TRICHOMONAS VAGINALIS, GIARDIA LAMBLIA, ENTAMOEBA HISTOLYTICA, TRYPANOSOMA BRUCEI GAMBIENSE
Chinedu
NECATOR AMERICANUS - New world hookworm ASCARIS LUMBRICOIDES- Ascarid TRICHURIS TRICHIURA- Whip worm
Chinedu
Hi guys
Chinedu
Why are myeloblasts not present in the bloodstream?
Chinedu
thank u
Dasaah
is blood from capillary puncture suitable for hormonal assays
LAFIA
In most cases the blood is necessary but it's limited from my own Analysis
Lee
u can do but it depends on your sampling blood its enev or not
Ali
Control indications of cipro, doxicyclin, setracyclin, cloxacyclin, erythromycin
oo
babinthe
Leaves accumulate on the forest floor at a rate of 2 g/cm2/yr and also decompose at a rate of 90% per year. Write a differential equation governing the number of grams of leaf litter per square centimeter of forest floor, assuming at time 0 there is no leaf litter on the ground. Does this amount approach a steady value? What is that value?
You have a cup of coffee at temperature 70°C, which you let cool 10 minutes before you pour in the same amount of milk at 1°C as in the preceding problem. How does the temperature compare to the previous cup after 10 minutes?
Abdul