▶️Series (Calculus)

►Jump back to previous note: Series (High school level)

Explicit Sequence vs. Recursive Sequence:

Explicit sequence would be presented as: a𝓃 = a₁ · kⁿ⁻¹. Recursive sequence would be presented as: a₁ = 3, a𝓃 = k · a𝓃₋₁

Sequence vs. Series:

Sequence is a LIST of numbers, Series is a NUMBER: the SUM of a sequence.

Convergence vs. Divergence:

Convergence means the limit of a function EXISTS. Divergence means the limit DOES NOT EXISTS.

Geometric Series in 𝚺 Notation

▼Refer to Cool Math: Geometric Series

Example

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Infinite Sequence (convergence | divergence)

►Jump to practice: Sequence convergence/divergence

Example

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• Easiest way: Apply the `L'hopital's Rule, take both Top's & Bottom's derivatives until both of them become numbers.

• So we get: 1/3.

Finite Geometric Series

►Jump to practice: Finite geometric series

Example

Solve:

• By using the Geometric Series formula, we get the informations as below:

  • Common ratio: r = -2

  • Amount of items: n = 20. Because k starts from 0, so there're 20 terms.

  • Initial term: a₀ = -4

• We calculate and get the result as below:

  

Partial Sums

Partial sums is just a fancy word for Finite series, because it's a a part of infinite series.

Example

Solve:

• The tricky part is how to count the amount of terms.

• Since n starts from 1, so there're 11 terms, which means we're to calculate S₁₁.

• S₁₁ = 88/16 = 11/2

Example

Solve:

• The tricky here is that: a𝓃 = S𝓃 - S𝓃-1, because S𝓃 = a₁ + a₂ + a₃ +.... + a𝓃-1 + a𝓃.

• So the result is: