Question and Answers Forum

All Questions      Topic List

Coordinate Geometry Questions

Previous in All Question      Next in All Question      

Previous in Coordinate Geometry      Next in Coordinate Geometry      

Question Number 44527 by Necxx last updated on 30/Sep/18

Commented by Necxx last updated on 30/Sep/18

21 please

$$\mathrm{21}\:{please} \\ $$

Commented by maxmathsup by imad last updated on 30/Sep/18

21) let S_n =Σ_(k=0) ^n  (−1)^k  C_n ^k     ((1+k λ)/((1+nλ)^k ))   with λ =ln(10)  S_n = Σ_(k=0) ^n    C_n ^k  (((−1)^k )/((1+nλ)^k )) +λ Σ_(k=0) ^n  C_n ^k (−1)^k   (k/((1+nλ)^k )) but  Σ_(k=0) ^n   C_n ^k (((−1)/(1+nυ)))^k   =(1−(1/(1+nλ)))^n  =(((nλ)/(1+nλ)))^n   Σ_(k=0) ^n   C_n ^k   k(((−1)/(1+nλ)))^k   ?  let find  Σ_(k=0) ^n  C_n ^k  kx^k      we have Σ_(k=0) ^n  x^k   =((x^(n+1) −1)/(x−1)) ⇒Σ_(k=1) ^n  k x^(k−1)  =((nx^(n+1)  −(n+1)x^n +1)/((1−x)^2 ))⇒  Σ_(k=1) ^n  k x^k  =(x/((1−x)^2 )){ nx^(n+1)  −(n+1)x^n  +1} ⇒  Σ_(k=0) ^n   C_n ^k   k (((−1)/(1+nλ)))^k  = ((−1)/((1+nλ)(1+(1/(1+nλ)))^2 )){ n(((−1)/(1+nλ)))^(n+1)  −(n+1)(((−1)/(1+nλ)))^n  +1}  rest to finich the calculus this is the way...

$$\left.\mathrm{21}\right)\:{let}\:{S}_{{n}} =\sum_{{k}=\mathrm{0}} ^{{n}} \:\left(−\mathrm{1}\right)^{{k}} \:{C}_{{n}} ^{{k}} \:\:\:\:\frac{\mathrm{1}+{k}\:\lambda}{\left(\mathrm{1}+{n}\lambda\right)^{{k}} }\:\:\:{with}\:\lambda\:={ln}\left(\mathrm{10}\right) \\ $$$${S}_{{n}} =\:\sum_{{k}=\mathrm{0}} ^{{n}} \:\:\:{C}_{{n}} ^{{k}} \:\frac{\left(−\mathrm{1}\right)^{{k}} }{\left(\mathrm{1}+{n}\lambda\right)^{{k}} }\:+\lambda\:\sum_{{k}=\mathrm{0}} ^{{n}} \:{C}_{{n}} ^{{k}} \left(−\mathrm{1}\right)^{{k}} \:\:\frac{{k}}{\left(\mathrm{1}+{n}\lambda\right)^{{k}} }\:{but} \\ $$$$\sum_{{k}=\mathrm{0}} ^{{n}} \:\:{C}_{{n}} ^{{k}} \left(\frac{−\mathrm{1}}{\mathrm{1}+{n}\upsilon}\right)^{{k}} \:\:=\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{1}+{n}\lambda}\right)^{{n}} \:=\left(\frac{{n}\lambda}{\mathrm{1}+{n}\lambda}\right)^{{n}} \\ $$$$\sum_{{k}=\mathrm{0}} ^{{n}} \:\:{C}_{{n}} ^{{k}} \:\:{k}\left(\frac{−\mathrm{1}}{\mathrm{1}+{n}\lambda}\right)^{{k}} \:\:?\:\:{let}\:{find}\:\:\sum_{{k}=\mathrm{0}} ^{{n}} \:{C}_{{n}} ^{{k}} \:{kx}^{{k}} \:\:\: \\ $$$${we}\:{have}\:\sum_{{k}=\mathrm{0}} ^{{n}} \:{x}^{{k}} \:\:=\frac{{x}^{{n}+\mathrm{1}} −\mathrm{1}}{{x}−\mathrm{1}}\:\Rightarrow\sum_{{k}=\mathrm{1}} ^{{n}} \:{k}\:{x}^{{k}−\mathrm{1}} \:=\frac{{nx}^{{n}+\mathrm{1}} \:−\left({n}+\mathrm{1}\right){x}^{{n}} +\mathrm{1}}{\left(\mathrm{1}−{x}\right)^{\mathrm{2}} }\Rightarrow \\ $$$$\sum_{{k}=\mathrm{1}} ^{{n}} \:{k}\:{x}^{{k}} \:=\frac{{x}}{\left(\mathrm{1}−{x}\right)^{\mathrm{2}} }\left\{\:{nx}^{{n}+\mathrm{1}} \:−\left({n}+\mathrm{1}\right){x}^{{n}} \:+\mathrm{1}\right\}\:\Rightarrow \\ $$$$\sum_{{k}=\mathrm{0}} ^{{n}} \:\:{C}_{{n}} ^{{k}} \:\:{k}\:\left(\frac{−\mathrm{1}}{\mathrm{1}+{n}\lambda}\right)^{{k}} \:=\:\frac{−\mathrm{1}}{\left(\mathrm{1}+{n}\lambda\right)\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{1}+{n}\lambda}\right)^{\mathrm{2}} }\left\{\:{n}\left(\frac{−\mathrm{1}}{\mathrm{1}+{n}\lambda}\right)^{{n}+\mathrm{1}} \:−\left({n}+\mathrm{1}\right)\left(\frac{−\mathrm{1}}{\mathrm{1}+{n}\lambda}\right)^{{n}} \:+\mathrm{1}\right\} \\ $$$${rest}\:{to}\:{finich}\:{the}\:{calculus}\:{this}\:{is}\:{the}\:{way}... \\ $$$$ \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com