The Fibonacci sequence was defined in Section 11.1 by the...

The Fibonacci sequence was defined in Section 11.1 by the equations $$f_{1}=1, \quad f_{2}=1, \quad f_{n}=f_{n-1}+f_{n-2} \quad n \geqslant 3$$ Show that each of the following statements is true. \begin{equation} \begin{array}{l}{\text { (a) } \frac{1}{f_{n-1} f_{n+1}}=\frac{1}{f_{n-1} f_{n}}-\frac{1}{f_{n} f_{n+1}}} \\ {\text { (b) } \sum_{n=2}^{\infty} \frac{1}{f_{n-1} f_{n+1}}=1 \quad \text { (c) } \sum_{n=2}^{\infty} \frac{f_{n}}{f_{n-1} f_{n+1}}=2}\end{array} \end{equation}

The Fibonacci sequence was defined in Section 11.1 by the
equations
$$f_{1}=1, \quad f_{2}=1, \quad f_{n}=f_{n-1}+f_{n-2} \quad n \geqslant 3$$
Show that each of the following statements is true.
\begin{equation}
\begin{array}{l}{\text { (a) } \frac{1}{f_{n-1} f_{n+1}}=\frac{1}{f_{n-1} f_{n}}-\frac{1}{f_{n} f_{n+1}}} \\ {\text { (b) } \sum_{n=2}^{\infty} \frac{1}{f_{n-1} f_{n+1}}=1 \quad \text { (c) } \sum_{n=2}^{\infty} \frac{f_{n}}{f_{n-1} f_{n+1}}=2}\end{array}
\end{equation}

Key Concepts

Mathematical Induction

Mathematical induction is a proof technique used to establish that a given statement holds for all natural numbers. It is especially useful in verifying identities or properties involving recursively defined sequences, such as those involving the Fibonacci sequence, where one proves a base case and then shows that if the statement holds for an arbitrary case, it holds for the next one.

Series Convergence

Series convergence is the concept dealing with whether an infinite series approaches a finite limit. In the context of summing series related to sequences like the Fibonacci numbers, determining convergence is crucial to ensure the sum is well-defined, and various tests and techniques can be applied to justify the convergence of series.

Recurrence Relations

Recurrence relations are equations that define sequences recursively. They form the backbone of sequences like the Fibonacci sequence, allowing the derivation of properties and identities by expressing terms in relation to their predecessors. Understanding recurrence relations is essential for analyzing the behavior and patterns within sequences.

Fibonacci Sequence

The Fibonacci sequence is a classical example of a recursively defined sequence where each term is the sum of the two preceding ones. This sequence is fundamental in various areas of mathematics and its properties, such as growth rate and relationships between terms, are often used to illustrate concepts in recursive formulas and series.

Telescoping Series

A telescoping series is one in which consecutive terms cancel each other out when summed, simplifying the evaluation of the sum, often to a small number of remaining terms. Recognizing a telescoping structure is a powerful technique for evaluating series, particularly when dealing with sequences defined by recursive relationships like the Fibonacci numbers.

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