Problem 5 - Calculus: Early Transcendentals

Question

Calculate the first eight terms of the sequence of partial sums correct to four decimal places. Does it appear that the series is convergent or divergent? \( \displaystyle \sum_{n = 1}^{\infty} \frac {1}{n^4 + n^2} \)

Step-by-Step Solution

Answer

The series is convergent.

Step 1: Understand the Sequence

The sequence given is \( a_n = \frac{1}{n^4 + n^2} \). We need to find the first eight terms of the sequence of partial sums, where each term of the sequence is defined as the reciprocal of \( n^4 + n^2 \).

Step 2: Calculate the First Eight Terms of the Sequence

Calculate each term \( a_n \) up to \( n = 8 \):- \( a_1 = \frac{1}{1^4 + 1^2} = \frac{1}{2} \)- \( a_2 = \frac{1}{2^4 + 2^2} = \frac{1}{20} \)- \( a_3 = \frac{1}{3^4 + 3^2} = \frac{1}{90} \)- \( a_4 = \frac{1}{4^4 + 4^2} = \frac{1}{272} \)- \( a_5 = \frac{1}{5^4 + 5^2} = \frac{1}{650} \)- \( a_6 = \frac{1}{6^4 + 6^2} = \frac{1}{1296} \)- \( a_7 = \frac{1}{7^4 + 7^2} = \frac{1}{2352} \)- \( a_8 = \frac{1}{8^4 + 8^2} = \frac{1}{4160} \)

Step 3: Compute Partial Sums

The partial sum \( S_n \) is defined as the sum of the first \( n \) terms of the sequence. Calculate each partial sum:- \( S_1 = a_1 = 0.5 \)- \( S_2 = a_1 + a_2 = 0.5 + 0.05 = 0.55 \)- \( S_3 = S_2 + a_3 = 0.55 + 0.0111 = 0.5611 \)- \( S_4 = S_3 + a_4 = 0.5611 + 0.00368 = 0.5648 \)- \( S_5 = S_4 + a_5 = 0.5648 + 0.00154 = 0.56634 \)- \( S_6 = S_5 + a_6 = 0.56634 + 0.000771 = 0.56711 \)- \( S_7 = S_6 + a_7 = 0.56711 + 0.000425 = 0.567535 \)- \( S_8 = S_7 + a_8 = 0.567535 + 0.000240 = 0.567775 \)

Step 4: Analyze Convergence

Examine the sequence of partial sums \( S_n \) obtained:- \( S_1 = 0.5 \)- \( S_2 = 0.55 \)- \( S_3 = 0.5611 \)- \( S_4 = 0.5648 \)- \( S_5 = 0.56634 \)- \( S_6 = 0.56711 \)- \( S_7 = 0.567535 \)- \( S_8 = 0.567775 \)The partial sums approach a limit, indicating that the series is convergent.

Key Concepts

Partial Sums

To understand series convergence, it's essential to grasp the concept of partial sums. A partial sum is the sum of the first 'n' terms of a sequence.
For instance, in the given sequence, each term is calculated, and these terms are added together to find the respective partial sums, denoted as \( S_n \).
If the partial sums \( S_1, S_2, S_3,... \) approach a specific value (a limit) as 'n' increases, the series can be said to converge.

• The partial sum \( S_1 = 0.5 \) comes from only the first term.
• Step by step, additional terms are added to get \( S_2 = 0.55 \), \( S_3 = 0.5611 \) and so on.
• The regular increase but in smaller increments implies a tendency towards a fixed number.

The progressive calculation of each partial sum helps you observe this steady approach to a limit.

Sequence of Terms

A sequence of terms is a list of numbers in a specific order, generated by a particular formula.
In the original exercise, the sequence is defined by the function \( a_n = \frac{1}{n^4 + n^2} \).
Here's how the sequence behaves:

• The sequence starts at \( a_1 = \frac{1}{2} \) and continues as \( a_2 = \frac{1}{20} \), \( a_3 = \frac{1}{90} \), etc.
• Each term is smaller than the previous, indicating that as 'n' grows, terms decrease significantly in size.

The behavior of these terms suggests that each new term adds less value to the overall sum, an indicator of potential convergence.

Convergent Series

A series is said to be convergent if its sequence of partial sums approaches a definite limit.
In other words, the series does not go towards infinity but settles down at a finite number.
From the solution, once you calculate several partial sums, like \( S_1 \) to \( S_8 \), you notice the numbers are getting closer and more consistent with each step:

• For example, \( S_8 = 0.567775 \) shows a very tiny increase from \( S_7 = 0.567535 \).
• This behavior typically shows that the series is converging, suggesting the perceived limit around the values being analyzed.

The concept of convergence is vital, especially in calculus, as it implies a stable and predictable output as more terms are added.

Calculus

Calculus is a field of mathematics focusing on change and motion. It heavily uses concepts such as limits, derivatives, and integrals.
When discussing series convergence, calculus provides tools and methods to determine if a series will converge or diverge.

• Limits play a crucial role as they help in identifying the behavior of a series.
• By calculating partial sums and analyzing them, we apply fundamental calculus principles.

The theory ensures sequences and series are manageable and understandable, predicting outcomes even as variables grow large. This understanding is foundational in many real-world applications, where predicting consistent behavior over infinite sequences is necessary.