Understanding Binomial Series with Binomial Coefficient/k Type

The binomial series is an algebraic series that expands expressions of the form $(1 + x)^n$, where $n$ is any real number, and $x$ is a real number such that $|x| < 1$ if $n$ is not a positive integer. When $n$ is a positive integer, the series is finite and is known as the binomial theorem. However, when $n$ is not a positive integer, the series is infinite.

Binomial Coefficient

The binomial coefficient is a fundamental component of the binomial theorem. It is denoted by $\binom{n}{k}$, read as "n choose k," and is defined as:

$$ \binom{n}{k} = \frac{n!}{k!(n-k)!} $$

where $n!$ denotes the factorial of $n$, and it is only defined for non-negative integers $k$ and $n$ with $k \leq n$.

Binomial Theorem

The binomial theorem states that for any non-negative integer $n$, the expansion of $(1 + x)^n$ is given by:

$$ (1 + x)^n = \sum_{k=0}^{n} \binom{n}{k} x^k $$

where the sum is taken over all non-negative integers $k$ from $0$ to $n$.

Binomial Series with Binomial Coefficient/k Type

When we introduce a division by $k$ into the binomial coefficient, we modify the standard binomial series. This can be seen in series expansions where each term has a factor of the form $\binom{n}{k}/k$. The modified series takes the form:

$$ \sum_{k=1}^{n} \frac{\binom{n}{k}}{k} x^k $$

Note that the summation starts from $k=1$ because $\binom{n}{0}/0$ is undefined.

Table of Differences and Important Points

Aspect	Standard Binomial Series	Binomial Series with Binomial Coefficient/k
Definition	$\sum_{k=0}^{n} \binom{n}{k} x^k$	$\sum_{k=1}^{n} \frac{\binom{n}{k}}{k} x^k$
Range of $k$	$0 \leq k \leq n$	$1 \leq k \leq n$
Division by $k$	No division by $k$	Each term includes division by $k$
Starting Index	$k = 0$	$k = 1$
Undefined Terms	No undefined terms	$\binom{n}{0}/0$ is undefined
Applicability	For non-negative integer $n$	For non-negative integer $n$
Factorial in Denominator	$k!(n-k)!$	$k \cdot k!(n-k)!$

Examples

Let's look at some examples to understand the binomial series with binomial coefficient/k type.

Example 1: Expansion of $(1 + x)^4$ with Binomial Coefficient/k

Using the standard binomial theorem, we have:

$$ (1 + x)^4 = \binom{4}{0} x^0 + \binom{4}{1} x^1 + \binom{4}{2} x^2 + \binom{4}{3} x^3 + \binom{4}{4} x^4 $$

Now, let's modify this with the binomial coefficient/k type:

$$ \sum_{k=1}^{4} \frac{\binom{4}{k}}{k} x^k = \frac{\binom{4}{1}}{1} x^1 + \frac{\binom{4}{2}}{2} x^2 + \frac{\binom{4}{3}}{3} x^3 + \frac{\binom{4}{4}}{4} x^4 $$

Simplifying the coefficients, we get:

$$ = 4x + 3x^2 + \frac{4}{3}x^3 + x^4 $$

Example 2: Expansion of $(1 + x)^3$ with Binomial Coefficient/k

Using the standard binomial theorem, we have:

$$ (1 + x)^3 = \binom{3}{0} x^0 + \binom{3}{1} x^1 + \binom{3}{2} x^2 + \binom{3}{3} x^3 $$

Now, let's modify this with the binomial coefficient/k type:

$$ \sum_{k=1}^{3} \frac{\binom{3}{k}}{k} x^k = \frac{\binom{3}{1}}{1} x^1 + \frac{\binom{3}{2}}{2} x^2 + \frac{\binom{3}{3}}{3} x^3 $$

Simplifying the coefficients, we get:

$$ = 3x + \frac{3}{2}x^2 + x^3 $$

In conclusion, the binomial series with binomial coefficient/k type is a variation of the standard binomial series that includes a division by the index $k$ in each term. This series starts from $k=1$ to avoid undefined terms and modifies the coefficients accordingly. It is important to note that this type of series is less common in standard mathematical texts but can arise in certain combinatorial or analytic contexts.