Illustration of the Sum Rule for Sequences
Consider two sequences \(a_n\) and \(b_n\), defined as:
- \(a_n = \frac{1}{n}\)
- \(b_n = \frac{2}{n}\)
We want to find the limit of the sum of these sequences as \(n\) approaches infinity:
\(\lim_{{n \to \infty}} (a_n + b_n) = \lim_{{n \to \infty}} \left(\frac{1}{n} + \frac{2}{n}\right)\)
Using the sum rule, we can break this down into the sum of the individual limits:
\(\lim_{{n \to \infty}} (a_n + b_n) = \lim_{{n \to \infty}} a_n + \lim_{{n \to \infty}} b_n\)
As \(n\) approaches infinity, both \(\frac{1}{n}\) and \(\frac{2}{n}\) approach 0:
\(\lim_{{n \to \infty}} a_n = 0\)
\(\lim_{{n \to \infty}} b_n = 0\)
So the sum of the limits is:
\(\lim_{{n \to \infty}} (a_n + b_n) = 0 + 0 = 0\)
This illustrates the sum rule for sequences, showing that the limit of the sum of two sequences is equal to the sum of the limits of the individual sequences.
Illustration of the Difference Rule for Sequences
Consider two sequences \(a_n\) and \(b_n\), defined as:
- \(a_n = \frac{1}{n}\)
- \(b_n = \frac{2}{n}\)
We want to find the limit of the difference of these sequences as \(n\) approaches infinity:
\(\lim_{{n \to \infty}} (a_n – b_n) = \lim_{{n \to \infty}} \left(\frac{1}{n} – \frac{2}{n}\right)\)
Using the difference rule, we can break this down into the difference of the individual limits:
\(\lim_{{n \to \infty}} (a_n – b_n) = \lim_{{n \to \infty}} a_n – \lim_{{n \to \infty}} b_n\)
As \(n\) approaches infinity, both \(\frac{1}{n}\) and \(\frac{2}{n}\) approach 0:
\(\lim_{{n \to \infty}} a_n = 0\)
\(\lim_{{n \to \infty}} b_n = 0\)
So the difference of the limits is:
\(\lim_{{n \to \infty}} (a_n – b_n) = 0 – 0 = 0\)
This illustrates the difference rule for sequences, showing that the limit of the difference of two sequences is equal to the difference of the limits of the individual sequences.
Illustration of the Product Rule for Sequences
Consider two sequences \(c_n\) and \(d_n\), defined as:
- \(c_n = \frac{1}{n}\)
- \(d_n = \frac{3}{n}\)
We want to find the limit of the product of these sequences as \(n\) approaches infinity:
\(\lim_{{n \to \infty}} (c_n \cdot d_n) = \lim_{{n \to \infty}} \left(\frac{1}{n} \cdot \frac{3}{n}\right)\)
Using the product rule, we can break this down into the product of the individual limits:
\(\lim_{{n \to \infty}} (c_n \cdot d_n) = \lim_{{n \to \infty}} c_n \cdot \lim_{{n \to \infty}} d_n\)
As \(n\) approaches infinity, both \(\frac{1}{n}\) and \(\frac{3}{n}\) approach 0:
\(\lim_{{n \to \infty}} c_n = 0\)
\(\lim_{{n \to \infty}} d_n = 0\)
So the product of the limits is:
\(\lim_{{n \to \infty}} (c_n \cdot d_n) = 0 \cdot 0 = 0\)
This illustrates the product rule for sequences, showing that the limit of the product of two sequences is equal to the product of the limits of the individual sequences.
Illustration of the Quotient Rule for Sequences
Consider two sequences \(e_n\) and \(f_n\), defined as:
- \(e_n = \frac{2}{n}\)
- \(f_n = \frac{4}{n^2}\)
We want to find the limit of the quotient of these sequences as \(n\) approaches infinity:
\(\lim_{{n \to \infty}} \frac{e_n}{f_n} = \lim_{{n \to \infty}} \frac{\frac{2}{n}}{\frac{4}{n^2}}\)
Using the quotient rule, we can break this down into the quotient of the individual limits:
\(\lim_{{n \to \infty}} \frac{e_n}{f_n} = \frac{\lim_{{n \to \infty}} e_n}{\lim_{{n \to \infty}} f_n}\)
As \(n\) approaches infinity, \(\frac{2}{n}\) approaches 0 and \(\frac{4}{n^2}\) approaches 0:
\(\lim_{{n \to \infty}} e_n = 0\)
\(\lim_{{n \to \infty}} f_n = 0\)
So the quotient of the limits is undefined, as we cannot divide by 0.
Summary: The quotient rule for sequences states that the limit of the quotient of two sequences is equal to the quotient of the limits of the individual sequences, provided that the limit of the denominator sequence is not zero.
Calculations: \(\lim_{{n \to \infty}} \frac{e_n}{f_n} = \frac{\lim_{{n \to \infty}} \frac{2}{n}}{\lim_{{n \to \infty}} \frac{4}{n^2}} = \frac{0}{0} = \text{undefined}\)
Illustration of the Constant Multiple Rule for Sequences
Consider a sequence \(g_n\) defined as:
- \(g_n = \frac{1}{n}\)
And let \(c = 3\), a constant value.
We want to find the limit of the constant multiple of the sequence as \(n\) approaches infinity:
\(\lim_{{n \to \infty}} (c \cdot g_n) = \lim_{{n \to \infty}} \left(3 \cdot \frac{1}{n}\right)\)
Using the constant multiple rule, we can take the constant out of the limit:
\(\lim_{{n \to \infty}} (c \cdot g_n) = c \cdot \lim_{{n \to \infty}} g_n\)
As \(n\) approaches infinity, \(\frac{1}{n}\) approaches 0:
\(\lim_{{n \to \infty}} g_n = 0\)
So the limit of the constant multiple is:
\(\lim_{{n \to \infty}} (c \cdot g_n) = 3 \cdot 0 = 0\)
Summary: The constant multiple rule for sequences states that the limit of a constant multiple of a sequence is equal to the constant times the limit of the sequence itself.
Calculations: \(\lim_{{n \to \infty}} (3 \cdot \frac{1}{n}) = 3 \cdot \lim_{{n \to \infty}} \frac{1}{n} = 3 \cdot 0 = 0\)
Illustration of the Power Rule for Sequences
Consider a sequence \( g_n \), defined as:
- \( g_n = \frac{1}{n} \)
We want to find the limit of this sequence raised to the power of 3 as \( n \) approaches infinity:
\( \lim_{{n \to \infty}} g_n^3 = \lim_{{n \to \infty}} \left(\frac{1}{n}\right)^3 \)
Using the power rule, we can find the limit of the sequence itself and then raise that limit to the power of 3:
\( \lim_{{n \to \infty}} g_n^3 = \left(\lim_{{n \to \infty}} g_n\right)^3 \)
As \( n \) approaches infinity, \( \frac{1}{n} \) approaches 0:
\( \lim_{{n \to \infty}} g_n = 0 \)
So the limit of the sequence raised to the power of 3 is 0:
\( \lim_{{n \to \infty}} g_n^3 = 0^3 = 0 \)
Summary: The power rule for sequences states that the limit of a sequence raised to the power of \( k \) is equal to the limit of the sequence itself raised to the power of \( k \).
Calculations: \( \lim_{{n \to \infty}} g_n^3 = \left(\lim_{{n \to \infty}} \frac{1}{n}\right)^3 = 0^3 = 0 \)
Illustration of the Reciprocal Power Rule for Sequences
Consider a sequence \( h_n \), defined as:
- \( h_n = n^2 \)
We want to find the limit of this sequence raised to the reciprocal of 2 (i.e., the square root) as \( n \) approaches infinity:
\( \lim_{{n \to \infty}} h_n^{1/2} = \lim_{{n \to \infty}} \sqrt{n^2} \)
Using the reciprocal power rule, we can find the limit of the sequence itself and then take the square root:
\( \lim_{{n \to \infty}} h_n^{1/2} = \sqrt{\lim_{{n \to \infty}} h_n} \)
As \( n \) approaches infinity, \( n^2 \) approaches infinity:
\( \lim_{{n \to \infty}} h_n = \infty \)
So the limit of the sequence raised to the reciprocal of 2 is infinity:
\( \lim_{{n \to \infty}} h_n^{1/2} = \sqrt{\infty} = \infty \)
Summary: The reciprocal power rule for sequences states that the limit of a sequence raised to the reciprocal of \( k \) is equal to the \( k \)-th root of the limit of the sequence itself.
Calculations: \( \lim_{{n \to \infty}} h_n^{1/2} = \sqrt{\lim_{{n \to \infty}} n^2} = \sqrt{\infty} = \infty \)
In mathematics, the rules of limits for sequences allow us to perform arithmetic operations on the limits of sequences. These rules are essential in understanding the behavior of sequences as they approach a specific value. The following are some examples that illustrate the application of these rules, considering two sequences \( a_n = \frac{1}{2} \) and \( b_n = \frac{1}{3} \), both as \( n \) approaches 5:
1. Sum Rule: \( \lim_{n \to 5} (a_n + b_n) = \lim_{n \to 5} \left( \frac{1}{2} + \frac{1}{3} \right) = \frac{1}{2} + \frac{1}{3} = \frac{5}{6} \)
2. Difference Rule: \( \lim_{n \to 5} (a_n – b_n) = \lim_{n \to 5} \left( \frac{1}{2} – \frac{1}{3} \right) = \frac{1}{2} – \frac{1}{3} = \frac{1}{6} \)
3. Constant Multiple Rule: \( \lim_{n \to 5} (3 \cdot a_n) = \lim_{n \to 5} \left( 3 \cdot \frac{1}{2} \right) = 3 \cdot \frac{1}{2} = \frac{3}{2} \)
4. Product Rule: \( \lim_{n \to 5} (a_n \cdot b_n) = \lim_{n \to 5} \left( \frac{1}{2} \cdot \frac{1}{3} \right) = \frac{1}{2} \cdot \frac{1}{3} = \frac{1}{6} \)
5. Quotient Rule: \( \lim_{n \to 5} \left( \frac{a_n}{b_n} \right) = \lim_{n \to 5} \left( \frac{\frac{1}{2}}{\frac{1}{3}} \right) = \frac{\frac{1}{2}}{\frac{1}{3}} = \frac{3}{2} \)