BeginnerTopic 21 of 104Engineering Mathematics
Sequences and series
Ordered lists of numbers and their sums; arithmetic and geometric series appear in finance and signal decay.
In depth
A sequence lists numbers; a series sums them. Arithmetic series add a constant difference, geometric series multiply by a constant ratio r with sum a/(1-r) when ‖r‖<1. Series model repeated payments, signal decay and approximations.
Key points
- Arithmetic sum = n(a₁+aₙ)/2
- Geometric sum = a(1-rⁿ)/(1-r)
- Convergence depends on ratio magnitude
ExampleA ball bouncing to 0.8 of its previous height has total travel governed by a geometric series.
Series describe accumulating effects, such as repeated payments, decaying signals and sums of harmonics, so they appear in finance and dynamics alike.
What is Sequences and series?
Ordered lists of numbers and their sums; arithmetic and geometric series appear in finance and signal decay.
Why is it important to understand sequences and series?
Series describe accumulating effects, such as repeated payments, decaying signals and sums of harmonics, so they appear in finance and dynamics alike.
How do I tell whether an infinite geometric series converges?
It converges when the absolute value of the common ratio is less than one, and its sum is the first term divided by one minus the ratio.
What is the difference between a sequence and a series?
A sequence is an ordered list of numbers. A series is the sum of the terms of a sequence.
How is Sequences and series used in practice?
Typical applications include estimating the total damage from a decaying vibration, summing the reflections in a transmission line and predicting the balance of a loan repaid in fixed instalments.
What should I learn before Sequences and series?
Study "Cross product" first (topic 20 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
What are the key points of Sequences and series?
Arithmetic sum = n(a₁+aₙ)/2. Geometric sum = a(1-rⁿ)/(1-r). Convergence depends on ratio magnitude.
Can you give an example of Sequences and series?
A ball bouncing to 0.8 of its previous height has total travel governed by a geometric series.
What comes after Sequences and series?
The next topic is "Complex numbers", which builds directly on this one.
How difficult is Sequences and series?
It is classed as Beginner in Engineering Mathematics (topic 21 of 104). It is an entry-level topic. It relies only on basic school mathematics and the ideas introduced earlier in this discipline.