Question

Recall that the Taylor series for \(f(x)=1 /(1-x)\) centered at 0 is the geometric series \(\sum_{i=0}^{\infty} x^{k}\) Show that this series can also be found as a binomial series.

Short answer

Yes, the Taylor series for the function \(f(x) = \frac{1}{1-x}\), which is a geometric series, can be found as a binomial series when considering the binomial expression \((1-x)^{-1}\). Both the geometric series and the binomial series are equivalent in this case.

Step-by-step solution

Binomial Series Formula

The binomial series for \((1 + x)^n\) is given by the infinite series: \begin{equation} (1+x)^n = \sum_{k=0}^{\infty} \binom{n}{k} x^k \end{equation} Where \(n\) is a real number and \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\).

Geometric Series Formula

The Taylor series for the function \(f(x) =\frac{1}{1-x}\) is given by the geometric series: \begin{equation} f(x) = \sum_{k=0}^{\infty} x^k \end{equation}

Connecting the Geometric and Binomial Series

To show that the geometric series can be found as a binomial series, we must find an expression for \((1+x)^n\) that matches with the given Taylor series of \(f(x)\). Since we want to match the geometric series \(\frac{1}{1-x}\), we should replace the x term in the binomial expression \((1+x)^n\) with an equivalent term that will result in the same geometric series. Let's consider \((1-x)^{-1}\). Applying the binomial formula to this expression we have: \begin{equation} (1-x)^{-1} = \sum_{k=0}^{\infty} \binom{-1}{k} (-x)^k \end{equation}

Simplifying the Expression

Now let's simplify the binomial coefficient and the series: \begin{equation} \binom{-1}{k} = \frac{(-1)!}{k!(-1-k)!} = \frac{1}{k!}\frac{(-1)(-2)\dots(-k)}{(-1)(-2)\dots(-k)}=(-1)^k \end{equation} Inserting this into the series we get: \begin{equation} (1-x)^{-1} = \sum_{k=0}^{\infty} (-1)^k (-x)^k = \sum_{k=0}^{\infty} x^k \end{equation}

Conclusion

We have demonstrated that the Taylor series for the function \(f(x)=\frac{1}{1-x}\), which is the geometric series \(\sum_{k=0}^{\infty} x^k\), can also be found as a binomial series when considering the binomial expression \((1-x)^{-1}\). In other words, the geometric series and the binomial series are equivalent which confirms the given statement.

Key concepts

Geometric Series

A geometric series is a fascinating mathematical concept that revolves around the idea of multiplying a fixed number, known as the 'common ratio', across terms in a sequence. This is a type of infinite series, written as \[ \sum_{k=0}^{\infty} ar^k \] where:

• \( a \) is the first term
• \( r \) is the common ratio

Whenever you evaluate a geometric series, you notice the sum of the sequence grows progressively in size—provided the ratio \( |r| < 1 \). In the context of Taylor series representations, such as those for \( f(x) = \frac{1}{1-x} \), the series becomes simply \[ \sum_{k=0}^{\infty} x^k \] showcasing a straightforward case where each successive term is multiplied by \( x \). This clean representation of power sequences makes the geometric series especially useful in mathematical calculations and analysis.

Binomial Series

The binomial series provides a powerful generalization of the concept of expanding expressions of the form \((1+x)^n\). This infinite series is applicable beyond just integers, as it is defined as: \[ (1 + x)^n = \sum_{k=0}^{\infty} \binom{n}{k} x^k \] It's useful across different mathematical domains, as it considers real numbers for \( n \). Differing from the finite expansion seen in typical binomial expressions for integer \( n \), when involving real numbers or even negatives, the expansion becomes infinite. This idea is central, for instance, in expressing complex functions as series, yielding insights into their behavior over specific ranges. The ability to apply this across values not fulfilling simple integer assumptions expands the application potential, especially in calculus and algebra.

Binomial Coefficient

A binomial coefficient defines the number of ways we can choose items from a collection, mathematically represented as \( \binom{n}{k} \). The formula for finding a binomial coefficient is \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \] where:

• \( n \) is the total number of items
• \( k \) is the number to choose
• \( ! \) represents a factorial

The simple essence of the binomial coefficient is captured in how it plays a critical role in expanding binomial expressions, forming an essential part of the binomial theorem. In series expressions, these coefficients provide the exact 'weight' each term needs, making them fundamental in ensuring the accurate representation of mathematical expansions that go into calculus and related applications.