Infinite Series
Sum of infinitely many terms.
Key idea
Sum of infinitely many terms.
In depth
An infinite series Σaₙ is the limit of partial sums Sₙ = a₁ + ... + aₙ. It converges if this limit exists. Even if terms tend to zero, the series may still diverge.
Worked example
Σ 1/2ⁿ = 1/2 + 1/4 + ... = 1. But Σ 1/n diverges even though 1/n → 0.
Formula
Real-world uses
- Modelling total payments of an infinite plan
- Adding up a bouncing ball’s distances
Common mistake
Thinking that aₙ → 0 implies convergence.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Infinite Series?
Sum of infinitely many terms. It is a topic at the expert level in Calculus.
What is the formula for Infinite Series?
The key formula is Σ aₙ.
Where is Infinite Series used in real life?
Real-world examples: Modelling total payments of an infinite plan; Adding up a bouncing ball’s distances.
What should I learn before Infinite Series?
Make sure you are comfortable with Sequences and Limits, the previous topic in Calculus.
What is a common mistake with Infinite Series?
Thinking that aₙ → 0 implies convergence.
How can I practise Infinite Series?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Infinite Series?
Move on to Geometric Series Convergence next.
Related expert topics in Calculus
- Derivatives of Inverse Trig – Derivatives of arcsin, arccos, arctan.
- Implicit Differentiation – Differentiate both sides of F(x,y)=0 treating y as a function of x.
- Logarithmic Differentiation – Take logs before differentiating products and powers.
- Related Rates – Link rates of change of connected quantities.
- Linear Approximation – Approximate f near a using its tangent line.
- Differentials – Use dy = f′(x)dx for small changes.