1

Use the following steps to prove $(17) .$ (a) Let $g(x)=\Sigma_{n=0}^{\infty}\left(\frac{k}{n}\right) x^{n} .$ Differentiate this series to show that $$g^{\prime}...

Question

Use the following steps to prove $(17) .$ (a) Let $g(x)=\Sigma_{n=0}^{\infty}\left(\frac{k}{n}\right) x^{n} .$ Differentiate this series to show that $$g^{\prime}(x)=\frac{k g(x)}{1+x} \quad-1< x<1$$ (b) Let $h(x)=(1+x)^{-k} g(x)$ and show that $h^{\prime}(x)=0$ (c) Deduce that $g(x)=(1+x)^{k}.$

Use the following steps to prove $(17) .$ (a) Let $g(x)=\Sigma_{n=0}^{\infty}\left(\frac{k}{n}\right) x^{n} .$ Differentiate this series to show that $$g^{\prime}(x)=\frac{k g(x)}{1+x} \quad-1< x<1$$ (b) Let $h(x)=(1+x)^{-k} g(x)$ and show that $h^{\prime}(x)=0$ (c) Deduce that $g(x)=(1+x)^{k}.$



Answers

Prove that the series obtained in Exercise 17 represents $ \sinh x $ for all $ x. $


Similar Solved Questions

5 answers
Consider the following_ f(x) = 7x - 1x2I(4)(a) Write the integral that describes the area of the shaded region:Tx(b) Find the area_Need Help?Read It_Watch_
Consider the following_ f(x) = 7x - 1x2 I(4) (a) Write the integral that describes the area of the shaded region: Tx (b) Find the area_ Need Help? Read It_ Watch_...
5 answers
NHbrans -(o)(g)(h)
NH brans - (o) (g) (h)...
5 answers
Gahinnrponenuz Louuliontaking Ue loganthmbobn -Hes Exerertsolubanlcuhmtcalciedodoamdl annrormalmeoutnn40jWnde ine culuon Intlermsnutura LJoarihnts ?The soklon sel & JUad iniudgtt decimals tor nuoa tla opaaalod Vanna declmal 4ularunanon Thmiontnta soltantceln (Npq nteger &1 dcumal rounded lo tao decmal paces neadad
Gahin nrponenuz Louulion taking Ue loganthm bobn -Hes Exerert soluban lcuhmt calciedo doamdl annrormalm eoutnn 40j Wnde ine culuon Intlerms nutura LJoarihnts ? The soklon sel & JUad iniudgtt decimals tor nuoa tla opaaalod Vanna declmal 4ularunanon Thmiont nta soltantceln (Npq nteger &1 dcuma...
5 answers
Answer al lhe point (49, point) Prevlous Find 5 Problem equation ot the Problem LIst une langent to Il
Answer al lhe point (49, point) Prevlous Find 5 Problem equation ot the Problem LIst une langent to Il...
5 answers
4. Show that PT<--S works by verifying that:Pr < -s[vJ; = [vJj andPr < -s[wJ;= [wlj 5. Construct the 4x4 matrix QS<-T
4. Show that PT<--S works by verifying that: Pr < -s[vJ; = [vJj and Pr < -s[wJ;= [wlj 5. Construct the 4x4 matrix QS<-T...
5 answers
1 the Use circle: polar coordinates 1 0! the 5 [41
1 the Use circle: polar coordinates 1 0! the 5 [ 4 1...
5 answers
For = lixed amount of E1s _lixed temperature_ what will happen if the volume is doubled?
For = lixed amount of E1s _ lixed temperature_ what will happen if the volume is doubled?...
5 answers
AcellusThe surface area of a postage stamp is 0.00600 m^2, and the air exerts 1.00 atm of pressure on it. How much force does it exert on the stamp? (Hint: The standard unit for pressure is Pa ) (Unit = N)
Acellus The surface area of a postage stamp is 0.00600 m^2, and the air exerts 1.00 atm of pressure on it. How much force does it exert on the stamp? (Hint: The standard unit for pressure is Pa ) (Unit = N)...
5 answers
A spring with spring constant of 33 N/m is stretched 0.16 m from its equilibrium position. How much work must be done to stretch it an additional 0.091 m?
A spring with spring constant of 33 N/m is stretched 0.16 m from its equilibrium position. How much work must be done to stretch it an additional 0.091 m?...
5 answers
Variation on a theme. Sedoheptulose 1,7 -bisphosphate is an intermediate in the Calvin cycle but not in the pentose phosphate pathway. What is the enzymatic basis of this difference?
Variation on a theme. Sedoheptulose 1,7 -bisphosphate is an intermediate in the Calvin cycle but not in the pentose phosphate pathway. What is the enzymatic basis of this difference?...
5 answers
5. [-/2 Points]DETAILSSCALCET8 12.4.019.Find two unit vectors orthogonal to both (7, 5, 1) and (-1, 1, 0) _(smaller i-value)(larger i-value)
5. [-/2 Points] DETAILS SCALCET8 12.4.019. Find two unit vectors orthogonal to both (7, 5, 1) and (-1, 1, 0) _ (smaller i-value) (larger i-value)...
5 answers
Write the equation of each line in general form.passes through points (3,5) and (-1,2)
Write the equation of each line in general form. passes through points (3,5) and (-1,2)...
5 answers
Use properties of limits and algebraic methods to find the limit, if it exists_ (If the limit is infinite, enter appropriate. If the limit does not otherwise exist, enter DNE: ) (11 If x < 2 lim_ f(x), where f(x) x2 Ifx 2 2
Use properties of limits and algebraic methods to find the limit, if it exists_ (If the limit is infinite, enter appropriate. If the limit does not otherwise exist, enter DNE: ) (11 If x < 2 lim_ f(x), where f(x) x2 Ifx 2 2...
5 answers
Fractal Geometry Draw the first four figures of the sequence described._____ is replaced by
Fractal Geometry Draw the first four figures of the sequence described. _____ is replaced by...
1 answers
An intense light source radiates uniformly in all directions. At a distance of 5.0$\mathrm { m }$ from the source, the radiation pressure on a perfectly absorbing surface is $9.0 \times 10 ^ { - 6 }$ Pa. What is the total average power output of the source?
An intense light source radiates uniformly in all directions. At a distance of 5.0$\mathrm { m }$ from the source, the radiation pressure on a perfectly absorbing surface is $9.0 \times 10 ^ { - 6 }$ Pa. What is the total average power output of the source?...
5 answers
Using the information in the table below, calculate theheat of released from the following reaction in kJ per g of NO. ( Mw. Of NO = 30.01 g/mol). 4NH3(g) + 5O2(g) 6H2O(g) + 4 NO(g)NH3H2ONOHfº (kJ/mol)- 46.1-241.8+ 90.3Select one:a. -8.372 b. -7.541c. -13.29 d. -5.678e. -30.16
Using the information in the table below, calculate the heat of released from the following reaction in kJ per g of NO. ( Mw. Of NO = 30.01 g/mol). 4NH3(g) + 5O2 (g) 6H2O(g) + 4 NO(g) NH3 H2O NO Hfº (kJ/mol) - 46.1 -241.8 + 90.3 Select one: a. -8.372 b. -7.541 c. -13.29 ...
5 answers
This contingency table describes 200 business students.MajorGenderAccounting (A)Economics (E)Statistics (S)Row TotalFemale (F)40252590Male (M)504020110Column Total906545200 Click here for the Excel Data FileIs major independent of gender?multiple choiceYesNo
This contingency table describes 200 business students. Major Gender Accounting (A) Economics (E) Statistics (S) Row Total Female (F) 40 25 25 90 Male (M) 50 40 20 110 Column Total 90 65 45 200 Click here for the Excel Data File Is major independent of gender? multiple choice Yes No...
5 answers
[SzCO]? (C is central)d. SeCN
[SzCO]? (C is central) d. SeCN...
5 answers
Aa entom Olivo conteclad blocks UaInU unolru hincl Rof mass 0 ko O8missstgand 0 0 iass 0*g} connccla ekopl on moteh aby an iaal ering Lhol pasaty horonta latzznk4 y barch utaa ovut en Eal Dultnt Jatan umutanu ELLEEenere Eha daaneuakh hrtach igWnat tho force of tension tne ideal string connecting and 0? Assume that the effect 0f @wr resistance negligible 136 N36 2 N 58.8 N22.6 N
Aa entom Olivo conteclad blocks UaInU unolru hincl Rof mass 0 ko O8missstgand 0 0 iass 0*g} connccla ekopl on moteh aby an iaal ering Lhol pasaty horonta latzznk4 y barch utaa ovut en Eal Dultnt Jatan umutanu ELLEEenere Eha daane uakh hrtach ig Wnat tho force of tension tne ideal string connecting ...

-- 0.022152--