Question

Induction 3. In class we discussed "telescoping series," meaning series of the forrm In class, we said that the partial sums are of the form The partial sums have a recursive characterization: s1bi -b and sn+1 Sa(0u+1-u+2). Use induction to show equation (1).

Answer 1

$\sum_{n=1}^{\infty}(b_n-b_{n+1})$

$S_1=b_1-b_2$ ----(2)

$S_{n+1}=S_n+(b_{n+1}-b_{n+2})$ ----(3)

n=1 (putting in equation(3))

$S_{2}=S_1+(b_{2}-b_{3})$

$S_{2}=(b_1-b_2)+(b_{2}-b_{3})$ (using value of S₁ from equation(2)

$S_{2}=b_1-b_{3}$

n=2

$S_{3}=S_2+(b_{3}-b_{4})$

(from above S₂=b₁-b₃)

n=3

$S_{4}=S_3+(b_{4}-b_{5})$

n=4

$S_{5}=S_4+(b_{5}-b_{6})$

$S_{5}=(b_1-b_5)+(b_{5}-b_{6})$

$S_{5}=b_1-b_{6}$

n=5

$S_{6}=S_5+(b_{6}-b_{7})$

$S_{6}=(b_1-b_6)+(b_{6}-b_{7})$

$S_{6}=b_1-b_{7}$

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So we can conclude from above

$S_{n}=b_1-b_{n+1}$