User:Egm4313.s12.team14.kimiagarov/HW 5

= Problem R5.1 =

Given

 * $$\displaystyle{1. \quad r(x) = \sum^\infin_{k=0}(k+1)kx^k}$$


 * $$\displaystyle{2. \quad r(x) = \sum^\infin_{k=0}{(-1)^k \over \gamma^k}x^{2k}}$$


 * $$\displaystyle{\gamma = constant}$$


 * $$\displaystyle{3. \quad \sin x\ at\ \hat x = 0}$$


 * $$\displaystyle{4. \quad \log (1+x)\ at\ \hat x = 0}$$


 * $$\displaystyle{5. \quad \log (1+x)\ at\ \hat x = 1}$$

Find

 * $$\displaystyle{R_c}$$

R5.1.1
$$\displaystyle{Using\ the\ ratio\ test;}$$

$$\displaystyle{\lim_{k \to \infty}\left| {a_{k+1} \over a_k} \right|<1}$$

$$\displaystyle{Where:}$$

$$\displaystyle{a_k = (k+1)kx^k}$$

$$\displaystyle{Plug\ Eqn.2\ into\ Eqn.1:}$$

$$\displaystyle{\lim_{k \to \infty}\left| {(k+2)(k+1)x^{k+1} \over (k+1)kx^k} \right|<1}$$

$$\displaystyle{\left| x \right|<1}$$

$$\displaystyle{R=1}$$

R5.1.2
$$\displaystyle{Using\ the\ ratio\ test;}$$

$$\displaystyle{\lim_{k \to \infty} \left| {a_{k+1} \over a_k} \right|}$$

$$\displaystyle{Where:}$$

$$\displaystyle{a_k ={(-1)^k \over \gamma^k}x^{2k}}$$

$$\displaystyle{\lim_{k \to \infty}\left| {{(-1)^{k+1} \over \gamma^{k+1}}x^{2(k+1)} \over {(-1)^k \over \gamma^k}x^{2k}} \right|<1}$$

$$\displaystyle{\left| {x^2 \over \gamma} \right|<1}$$

$$\displaystyle{\left| x \right| < \sqrt{ \left| \gamma  \right|}}$$

$$\displaystyle{R = \sqrt{ \left| \gamma \right|}}$$

R5.1.3
$$\displaystyle{Using\ Taylor\ series\ expansion;}$$

$$\displaystyle{\sum^\infin_{k=0}{f^{(n)}(\hat x) \over n!}(x- \hat x)^n\ at\ \hat x=0}$$

$$\displaystyle{{\sin0 \over 0!}x^0+{\cos0 \over 1!}x^1-{\sin0 \over 2!}x^2-{\cos0 \over 3!}x^3...}$$

$$\displaystyle{{1 \over 1!}x^1-{1 \over 3!}x^3 ...}$$

$$\displaystyle{\sum^\infin_{k=0}{(-1)^k \over (2k+1)!}x^{k+1}}$$

$$\displaystyle{Using\ (2)p.7-31;}$$

$$\displaystyle{R_c = \left[ \lim_{k \to \infty} \left| {d_{k+1} \over d_k} \right| \right]^{-1}}$$

$$\displaystyle{Where\ d_k = {(-1)^k \over (2k+1)!}}$$

$$\displaystyle{R_c = \left[ \lim_{k \to \infty} \left| {{(-1)^{k+1} \over (2(k+1)+1)!} \over {(-1)^k \over (2k+1)!}} \right| \right]^{-1}}$$

$$\displaystyle{R_c = \left[ \lim_{k \to \infty} \left| {{1 \over (2k+3)!} \over {1 \over (2k+1)!}} \right| \right]^{-1}}$$

$$\displaystyle{R_c = \left[ \lim_{k \to \infty} \left| {(2k+1)! \over (2k+3)(2k+2)(2k+1)!} \right| \right]^{-1}}$$

$$\displaystyle{R_c = \left[ \lim_{k \to \infty} \left| {1 \over (2k+3)(2k+2)} \right| \right]^{-1}}$$

$$\displaystyle{R_c = \left[ 0 \right]^{-1}}$$

$$\displaystyle{R_c = \infty}$$

R5.1.4
$$\displaystyle{Using\ Taylor\ series\ expansion;}$$

$$\displaystyle{\sum^\infin_{n=0}{f^{(n)}(\hat x) \over n!}(x- \hat x)^n\ at\ \hat x=0}$$

$$\displaystyle{{log(1+0) \over 0!}x^0+{(1+0)^{-1} \over 1!}x^1-{(1+0)^{-2} \over 2!}x^2+{2(1+0)^{-3} \over 3!}x^3-{6(1+0)^{-4} \over 4!}x^4 ...}$$

$$\displaystyle{0+x^1-{x^2 \over 2!}+{2x^3 \over 3!}-{6x^4 \over 4!} ...}$$

$$\displaystyle{\sum^\infin_{k=0}{(-1)^kx^{k+1} \over (k+1)}}$$

$$\displaystyle{Using\ (2)p.7-31;}$$

$$\displaystyle{R_c = \left[ \lim_{k \to \infty} \left| {d_{k+1} \over d_k} \right| \right]^{-1}}$$

$$\displaystyle{Where\ d_k = {(-1)^k \over (k+1)}}$$

$$\displaystyle{R_c = \left[ \lim_{k \to \infty} \left| {{(-1)^{k+1} \over (k+1+1)} \over {(-1)^k \over (k+1)}} \right| \right]^{-1}}$$

$$\displaystyle{R_c = \left[ \lim_{k \to \infty} \left| {{1 \over (k+2)} \over {1 \over (k+1)}} \right| \right]^{-1}}$$

$$\displaystyle{R_c = \left[ 1 \right]^{-1}}$$

$$\displaystyle{R_c = 1}$$

R5.1.5
$$\displaystyle{Using\ Taylor\ series\ expansion;}$$

$$\displaystyle{\sum^\infin_{n=0}{f^{(n)}(\hat x) \over n!}(x- \hat x)^n\ at\ \hat x=0}$$

$$\displaystyle{{log(1+1) \over 0!}(x-1)^0+{(1+1)^{-1} \over 2!}(x-1)^1-{(1+1)^{-2} \over 3!}(x-1)^2+{2(1+1)^{-3} \over 4!}(x-1)^3-{6(1+1)^{-4} \over 5!}(x-1)^4 ...}$$

$$\displaystyle{log(2)+(x-1)^1 2^{-1}-{(x-1)^2 2^{-2} \over 2!}+{2(x-1)^3 2^{-3} \over 3!}-{6(x-1)^4 2^{-4} \over 4!} ...}$$

$$\displaystyle{log(2) + \sum^\infin_{k=0}{(-1)^k(x-1)^{k+1} 2^{-(k+1)} \over (k+1)}}$$

$$\displaystyle{Note:\ log(2)\ will\ not\ affect\ R_c}$$

$$\displaystyle{Using\ (2)p.7-31;}$$

$$\displaystyle{R_c = \left[ \lim_{k \to \infty} \left| {d_{k+1} \over d_k} \right| \right]^{-1}}$$

$$\displaystyle{Where\ d_k = {(-1)^k 2^{-(k+1)} \over (k+1)}}$$

$$\displaystyle{R_c = \left[ \lim_{k \to \infty} \left| {{(-1)^{k+1} 2^{-(k+1+1)} \over (k+1+1)} \over {(-1)^k 2^{-(k+1)} \over (k+1)}} \right| \right]^{-1}}$$

$$\displaystyle{R_c = \left[ \lim_{k \to \infty} \left| \right| \right]^{-1}}$$

$$\displaystyle{R_c = \left[ \lim_{k \to \infty} \left| {(k+1) \over 2(k+2)} \right| \right]^{-1}}$$

$$\displaystyle{R_c = \left[ {1 \over 2} \right]^{-1}}$$

$$\displaystyle{R_c = 2}$$