3.2 Continuity, Derivatives, and Integrals 125Solution.Recall that the Fibonacci numbers satisfy the Cassini identityFn+ 1 Fn− 1 −Fn^2 =(− 1 )n.Hence∏∞n= 1(
1 +
(− 1 )n
Fn^2)
= lim
N→∞∏N
n= 1Fn^2 +(− 1 )n
Fn^2
= lim
N→∞∏N
n= 1Fn− 1
Fn·
Fn+ 1
Fn= lim
N→∞F 0 FN+ 1
F 1 FN
= lim
N→∞FN+ 1
FN
Because of the Binet formulaFn=1
√
5
⎡
⎣
(
1 +
√
5
2
)n+ 1
−(
1 −
√
5
2
)n+ 1 ⎤
⎦, forn≥ 0 ,the above limit is equal to^1 +√
5
2.
377.Compute the product
(
1 −4
1
)(
1 −
4
9
)(
1 −
4
25
)
···.
378.Letxbe a positive number less than 1. Compute the product∏∞n= 0(
1 +x^2
n)379.Letxbe a real number. Define the sequence(xn)n≥ 1 recursively byx 1 =1 and
xn+ 1 =xn+nxnforn≥1. Prove that∏∞n= 1(
1 −
xn
xn+ 1)
=e−x.3.2 Continuity, Derivatives, and Integrals
3.2.1 Limits of Functions.......................................
Among the various ways to find the limit of a function, the most basic is the definition
itself.