Lesson 3.1: Standard summation formulae

Introduction

Welcome to Lesson 3.1 on Standard Summation Formulae! 🎉 In this lesson, we will explore essential summation results that serve as building blocks for advanced mathematics. By the end of this lesson, you will be able to:

• Understand the formulas for $\Sigma r$, $\Sigma r^2$, and $\Sigma r^3$ and learn how to prove them using induction.
• Sum linear combinations of these standard series.
• Change limits and re-index sums.
• Apply the summation formulae in various contexts.
• Evaluate sums that are polynomials in $r$ in a closed form.

Let's dive in!

Standard Summation Results

Summation of Natural Numbers

The first summation we will explore is the sum of the first $n$ natural numbers. The formula is:

$$\Sigma_{r=1}^{n} r = \frac{n(n + 1)}{2}$$

Example:

If you want to find the sum of the first 5 natural numbers:

$$\Sigma_{r=1}^{5} r = 1 + 2 + 3 + 4 + 5 = 15$$

Using the formula:

$$\frac{5(5 + 1)}{2} = \frac{5 \cdot 6}{2} = 15$$

Summation of Squares

Next, we have the sum of the squares of the first $n$ natural numbers, given by:

$$\Sigma_{r=1}^{n} r^2 = \frac{n(n + 1)(2n + 1)}{6}$$

Example:

To compute the sum of the squares of the first 4 natural numbers:

$$\Sigma_{r=1}^{4} r^2 = 1^2 + 2^2 + 3^2 + 4^2 = 1 + 4 + 9 + 16 = 30$$

Using the formula:

$$\frac{4(4 + 1)(2 \cdot 4 + 1)}{6} = \frac{4 \cdot 5 \cdot 9}{6} = 30$$

Summation of Cubes

We also have the sum of the cubes of the first $n$ natural numbers:

$$\Sigma_{r=1}^{n} r^3 = \left(\frac{n(n + 1)}{2}

ight)^2$$

Example:

To find the sum of the cubes of the first 3 natural numbers:

$$\Sigma_{r=1}^{3} r^3 = 1^3 + 2^3 + 3^3 = 1 + 8 + 27 = 36$$

Using the formula:

$$\left(\frac{3(3 + 1)}{2}

$ight)^2 = \left(\frac{3 \cdot 4}{2}$

ight)^2 = (6)^2 = 36$$

Proof by Induction

To illustrate the validity of these summation formulas, we will prove the formula for $\Sigma r$ using mathematical induction.

$1. Base Case (n=1):$

$\Sigma_{r=1}^{1} r = 1 = \frac{1(1 + 1)}{2}$ ✓

1. Inductive Step:

Assume the formula holds for $n = k$:

$$\Sigma_{r=1}^{k} r = \frac{k(k + 1)}{2}$$

Now consider $n = k + 1$:

$$\Sigma_{r=1}^{k+1} r = \Sigma_{r=1}^k r + (k + 1)$$

By the inductive hypothesis:

$$= \frac{k(k + 1)}{2} + (k + 1)$$

$$= \frac{k(k + 1) + 2(k + 1)}{2}$$

Factor out $(k + 1)$:

$$= \frac{(k + 1)(k + 2)}{2}$$

Thus, the formula is proved!

Linear Combinations of Summations

When dealing with linear combinations, we can manipulate the sums easily. For example, consider the expression:

$$\Sigma_{r=1}^{n} (3r + 2)$$

This can be split into two separate summations:

$$3\Sigma_{r=1}^{n} r + 2\Sigma_{r=1}^{n} 1$$

Using our previous results:

$$3 \cdot \frac{n(n + 1)}{2} + 2n$$

Example:

Calculate $\Sigma_{r=1}^{4} (3r + 2)$:

$$= 3\Sigma_{r=1}^{4} r + 2 \cdot 4 = 3 \cdot 10 + 8 = 38$$

Changing Limits and Re-indexing

Changing limits in a summation can simplify problems significantly. If we have a sum that starts at $m$ instead of $1$, we can rewrite it as:

$$\Sigma_{r=m}^{n} f(r) = \Sigma_{r=1}^{n} f(r + m - 1)$$

Example:

If we need to evaluate:

$$\Sigma_{r=3}^{5} r$$

We can express it as:

$$\Sigma_{r=1}^{5} r - \Sigma_{r=1}^{2} r = 15 - 3 = 12$$

Conclusion

In this lesson, we covered important summation formulas and their applications. These formulas provide a basis for understanding more complex series, such as geometric and harmonic series, and they will be essential for future mathematical concepts.

Study Notes

• Summation formulas:
• $\Sigma_{r=1}^{n} r = \frac{n(n + 1)}{2}$
• $\Sigma_{r=1}^{n} r^2 = \frac{n(n + 1)(2n + 1)}{6}$
• $$\Sigma_{r=1}$^{n} r^3 = $\left($$\frac{n(n + 1)}{2}

ight)^2

• Understand and apply the method of induction to prove formulas.
• Be able to sum linear combinations of series.
• Change limits and re-index summations effectively.
• Evaluate polynomial sums in closed form.