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Question Number 12109 Answers: 0 Comments: 0

show that: Σ_(n=1) ^∞ (((−1)^(n+1) )/n)=ln(2) please show your working

$$\mathrm{show}\:\mathrm{that}: \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}+\mathrm{1}} }{{n}}=\mathrm{ln}\left(\mathrm{2}\right) \\ $$$$\mathrm{please}\:\mathrm{show}\:\mathrm{your}\:\mathrm{working} \\ $$

Question Number 12096 Answers: 1 Comments: 0

Question Number 12093 Answers: 1 Comments: 0

log_(abc) b=3 log_(abc) c=4 log_(abc) a=?

$$\mathrm{log}_{\mathrm{abc}} \mathrm{b}=\mathrm{3} \\ $$$$\mathrm{log}_{\mathrm{abc}} \mathrm{c}=\mathrm{4} \\ $$$$\mathrm{log}_{\mathrm{abc}} \mathrm{a}=? \\ $$

Question Number 12080 Answers: 1 Comments: 0

Question Number 12074 Answers: 1 Comments: 0

Question Number 12030 Answers: 1 Comments: 0

Question Number 12017 Answers: 0 Comments: 0

How Can we expand (a+b)^(1/2) and (a+b)^(−n) ?

$${How}\:{Can}\:{we}\:{expand}\:\left({a}+{b}\right)^{\frac{\mathrm{1}}{\mathrm{2}}} \:{and} \\ $$$$\left({a}+{b}\right)^{−{n}} \:? \\ $$

Question Number 11914 Answers: 0 Comments: 0

Turevlenebilir bir f fonksiyonu icin f(x+y)=f(x)+f(y)+2xy ve f′(0)=−3 old.gore f′(2)=? czm∵ f(x+y)=f(x)+f(y)+2xy y yi sabit kabul edersek f′(x+y)=f′(x)+2y x=0,y=2 icn f′(2)=f′(0)+2.2 f′(2)=−3+4=1

$${Turevlenebilir}\:{bir}\:{f}\:{fonksiyonu}\:{icin} \\ $$$${f}\left({x}+{y}\right)={f}\left({x}\right)+{f}\left({y}\right)+\mathrm{2}{xy}\:{ve}\:{f}'\left(\mathrm{0}\right)=−\mathrm{3} \\ $$$${old}.{gore}\:{f}'\left(\mathrm{2}\right)=? \\ $$$${czm}\because\:\:{f}\left({x}+{y}\right)={f}\left({x}\right)+{f}\left({y}\right)+\mathrm{2}{xy} \\ $$$${y}\:{yi}\:{sabit}\:{kabul}\:{edersek} \\ $$$${f}'\left({x}+{y}\right)={f}'\left({x}\right)+\mathrm{2}{y} \\ $$$${x}=\mathrm{0},{y}=\mathrm{2}\:{icn} \\ $$$${f}'\left(\mathrm{2}\right)={f}'\left(\mathrm{0}\right)+\mathrm{2}.\mathrm{2} \\ $$$${f}'\left(\mathrm{2}\right)=−\mathrm{3}+\mathrm{4}=\mathrm{1} \\ $$

Question Number 11901 Answers: 0 Comments: 0

((7cos^2 x+sin^2 x−3)/(2cos^2 x−sin^2 x))=? czm∵ ((7cos^2 x+1−cos^2 x−3)/(2cos^2 x−sin^2 x)) ((6cos^2 x−2)/(2cos^2 x−(1−cos^2 x)))=((6cos^2 x−2)/(3cos^2 x−1)) ((2(3cos^2 x−1))/(3cos^2 x−1))=2

Question Number 11869 Answers: 0 Comments: 0

Why the expansion of (a+b)^(−n) follows newton′s expansion rule?

$${Why}\:{the}\:{expansion}\:{of}\:\:\left({a}+{b}\right)^{−{n}} \:{follows} \\ $$$${newton}'{s}\:{expansion}\:{rule}? \\ $$

Question Number 11831 Answers: 0 Comments: 0

the system of equation a − (√(c^2 −(1/(16)) ))= (√(b^2 − (1/(16)))) b − (√(a^2 − (1/(25))))= (√(c^2 − (1/(25)))) c − (√(b^2 − (1/(36))))= (√(a^2 − (1/(36)))) given that a, b, c are real numbers that satisfy they system of equation ... if a + b + c = (x/(√(y ))) where x, y are positive integers and y is square free find the value of x + y

$$\mathrm{the}\:\mathrm{system}\:\mathrm{of}\:\mathrm{equation} \\ $$$$ \\ $$$$\mathrm{a}\:−\:\sqrt{\mathrm{c}^{\mathrm{2}} \:−\frac{\mathrm{1}}{\mathrm{16}}\:}=\:\sqrt{\mathrm{b}^{\mathrm{2}} \:−\:\frac{\mathrm{1}}{\mathrm{16}}} \\ $$$$\mathrm{b}\:−\:\sqrt{\mathrm{a}^{\mathrm{2}} \:−\:\frac{\mathrm{1}}{\mathrm{25}}}=\:\sqrt{\mathrm{c}^{\mathrm{2}} \:−\:\frac{\mathrm{1}}{\mathrm{25}}} \\ $$$$\mathrm{c}\:−\:\sqrt{\mathrm{b}^{\mathrm{2}} \:−\:\frac{\mathrm{1}}{\mathrm{36}}}=\:\sqrt{\mathrm{a}^{\mathrm{2}} \:−\:\frac{\mathrm{1}}{\mathrm{36}}} \\ $$$$ \\ $$$$\mathrm{given}\:\mathrm{that}\:\mathrm{a},\:\mathrm{b},\:\mathrm{c}\:\mathrm{are}\:\mathrm{real}\:\mathrm{numbers}\: \\ $$$$\mathrm{that}\:\mathrm{satisfy}\:\mathrm{they}\:\mathrm{system}\:\mathrm{of}\:\mathrm{equation}\:... \\ $$$$\mathrm{if}\:\mathrm{a}\:+\:\mathrm{b}\:+\:\mathrm{c}\:=\:\frac{\mathrm{x}}{\sqrt{\mathrm{y}\:}}\:\mathrm{where}\:\mathrm{x},\:\mathrm{y}\:\mathrm{are}\: \\ $$$$\mathrm{positive}\:\mathrm{integers}\:\mathrm{and}\:\mathrm{y}\:\mathrm{is}\:\mathrm{square}\:\mathrm{free} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{x}\:+\:\mathrm{y}\: \\ $$$$ \\ $$

Question Number 11813 Answers: 1 Comments: 3

Question Number 11805 Answers: 2 Comments: 0

x^y = y^x x^2 = y^3 find x and y

$$\mathrm{x}^{\mathrm{y}} \:=\:\mathrm{y}^{\mathrm{x}} \:\:\:\: \\ $$$$\mathrm{x}^{\mathrm{2}} \:=\:\mathrm{y}^{\mathrm{3}} \\ $$$$\mathrm{find}\:\mathrm{x}\:\mathrm{and}\:\mathrm{y} \\ $$

Question Number 11768 Answers: 0 Comments: 0

ax^3 +bx^2 +cx+d=0 solve it.

$${ax}^{\mathrm{3}} +{bx}^{\mathrm{2}} +{cx}+{d}=\mathrm{0}\:{solve}\:{it}. \\ $$

Question Number 11766 Answers: 1 Comments: 0

∫(1/(sin2x))dx=?

$$\int\frac{\mathrm{1}}{\mathrm{sin2x}}\mathrm{dx}=? \\ $$

Question Number 11764 Answers: 1 Comments: 0

if x y and z are solution of ((x + xy)/(x + y + 1)) = 2 ((2x + xz)/(x + z +2)) = 3 ((2 + 2y+ z + yz)/(y + z +3)) = 4 so, the value of (1/x) + (1/(y + 1)) + (1/(z + 2 )) = ....?

$$\mathrm{if}\:\mathrm{x}\:\mathrm{y}\:\mathrm{and}\:\mathrm{z}\:\mathrm{are}\:\mathrm{solution}\:\mathrm{of} \\ $$$$\frac{\mathrm{x}\:+\:\mathrm{xy}}{\mathrm{x}\:+\:\mathrm{y}\:+\:\mathrm{1}}\:=\:\mathrm{2} \\ $$$$\frac{\mathrm{2x}\:+\:\mathrm{xz}}{\mathrm{x}\:+\:\mathrm{z}\:+\mathrm{2}}\:=\:\mathrm{3} \\ $$$$\frac{\mathrm{2}\:+\:\mathrm{2y}+\:\mathrm{z}\:+\:\mathrm{yz}}{\mathrm{y}\:+\:\mathrm{z}\:+\mathrm{3}}\:=\:\mathrm{4} \\ $$$$\mathrm{so},\:\mathrm{the}\:\mathrm{value}\:\mathrm{of} \\ $$$$\frac{\mathrm{1}}{\mathrm{x}}\:+\:\frac{\mathrm{1}}{\mathrm{y}\:+\:\mathrm{1}}\:+\:\frac{\mathrm{1}}{\mathrm{z}\:+\:\mathrm{2}\:}\:=\:....? \\ $$$$ \\ $$

Question Number 11754 Answers: 1 Comments: 0

How many numbers between 1 − 2017 that aren′t divisible by 5, 6, 7, 8 ?

$$\mathrm{How}\:\mathrm{many}\:\mathrm{numbers}\:\mathrm{between}\:\mathrm{1}\:−\:\mathrm{2017} \\ $$$$\mathrm{that}\:\mathrm{aren}'\mathrm{t}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{5},\:\mathrm{6},\:\mathrm{7},\:\mathrm{8}\:? \\ $$

Question Number 11753 Answers: 2 Comments: 0

a, b, c are the roots from equation x^3 − 5x^2 − 9x + 10 = 0 If P(x) = Ax^3 + Bx^2 + Cx − 2015 and P(a) = b + c P(b) = a + c P(c) = a + b What is the value of A + B + C ?

$${a},\:{b},\:{c}\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{from}\:\mathrm{equation}\:{x}^{\mathrm{3}} \:−\:\mathrm{5}{x}^{\mathrm{2}} \:−\:\mathrm{9}{x}\:+\:\mathrm{10}\:=\:\mathrm{0} \\ $$$$\mathrm{If}\:{P}\left({x}\right)\:=\:{Ax}^{\mathrm{3}} \:+\:{Bx}^{\mathrm{2}} \:+\:{Cx}\:−\:\mathrm{2015}\:\mathrm{and} \\ $$$${P}\left({a}\right)\:=\:{b}\:+\:{c}\: \\ $$$${P}\left({b}\right)\:=\:{a}\:+\:{c} \\ $$$${P}\left({c}\right)\:=\:{a}\:+\:{b} \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{A}\:+\:{B}\:+\:{C}\:? \\ $$

Question Number 11748 Answers: 1 Comments: 0

Solve simultaneously x^2 + 4y^2 + xy = 6 .............. equation (i) 3x^2 + 8y^2 = 14 .............. equation (ii)

$$\mathrm{Solve}\:\mathrm{simultaneously} \\ $$$$\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{4y}^{\mathrm{2}} \:+\:\mathrm{xy}\:=\:\mathrm{6}\:\:\:\:\:..............\:\mathrm{equation}\:\left(\mathrm{i}\right) \\ $$$$\mathrm{3x}^{\mathrm{2}} \:+\:\mathrm{8y}^{\mathrm{2}} \:=\:\mathrm{14}\:\:\:\:\:\:\:\:\:\:\:..............\:\mathrm{equation}\:\left(\mathrm{ii}\right) \\ $$

Question Number 11741 Answers: 1 Comments: 0

∫xtan^2 x dx=?

$$\int\mathrm{xtan}^{\mathrm{2}} \:\mathrm{x}\:\mathrm{dx}=? \\ $$

Question Number 11714 Answers: 0 Comments: 3

Solve the Crazy equation... x(lnx)^2 +xlnx−1=0

$${Solve}\:{the}\:{Crazy}\:{equation}... \\ $$$${x}\left({lnx}\right)^{\mathrm{2}} +{xlnx}−\mathrm{1}=\mathrm{0} \\ $$

Question Number 11707 Answers: 0 Comments: 0

ax^3 +bx^2 +cx+d=0 pls. solve it in a alzebric way , that a O level(ssc) student can understand...