∞∑k=0(−1)k(k+14k+1)= ? Calculus Power Series Lagrange Form of the Remainder Term in a Taylor Series 1 Answer Cesareo R. Mar 3, 2017 See below. Explanation: 1(x+1)2=−ddx(1x+1) but 1x+1=12+x−1=12(11+x−12)=12∞∑k=0(−1)k(x−12)k for ∣∣∣x−12∣∣∣<1 but −ddx(12∞∑k=0(−1)k(x−12)k)=−122∞∑k=1(−1)kk2k−1(x−1)k−1=∞∑k=0(−1)k(k+12k+2)(x−1)k Making x=12 we have ∞∑k=0(−1)k(k+14k+1)=1(12+1)2=(23)2 Answer link Related questions What is the Lagrange Form of the Remainder Term in a Taylor Series? What is the Remainder Term in a Taylor Series? How do you find the Remainder term in Taylor Series? How do you find the remainder term R3(x;1) for f(x)=sin(2x)? How do you find the Taylor remainder term Rn(x;3) for f(x)=e4x? How do you find the Taylor remainder term R3(x;0) for f(x)=12+x? How do you use the Taylor Remainder term to estimate the error in approximating a function... How do you find the smallest value of n for which the Taylor Polynomial pn(x,c) to... How do you find the largest interval (c−r,c+r) on which the Taylor Polynomial pn(x,c)... How do you find the smallest value of n for which the Taylor series approximates the function... See all questions in Lagrange Form of the Remainder Term in a Taylor Series Impact of this question 2179 views around the world You can reuse this answer Creative Commons License