Using ➕partial sums➕ to find the nth term of the series 🤩🤩 #apcalculus #apcalc #unit10 #shorts

We're tackling a fundamental concept from Unit 10 of AP Calc: finding the nth term of a sequence using its partial sum sequences.

Consider a series a_n, where S represents the sum of all terms a_1+a_2+a_3+,... indefinitely. Now, imagine we're presented with a partial sum sequence, S_n. If asked to find a specific term in the series, say a_6, we look at the sum of the series up to that term. By understanding that S_6 is the sum of the first six terms and S_5 is the sum of the first five terms, we realize that the sixth term, a_6, can be found by subtracting S_5 from S_6.

This method, where a_n=S_n-S_(n-1), is a straightforward and efficient way to identify any term within a series, based on the sequence of its partial sums.

#APCalculus #MathEducation #CalculusConcepts #PartialSumSequences #NthTerm

Unit 10 of AP Calculus is all about Infinite Sequences and Series:
10.1 Defining Convergent and Divergent Infinite Series
10.2 Working with Geometric Series
10.3 The nth Term Test for Divergence
10.4 Integral Test for Convergence
10.5 Harmonic Series and p-Series
10.6 Comparison Tests for Convergence
10.7 Alternating Series Test for Convergence
10.8 Ratio Test for Convergence
10.9 Determining Absolute or Conditional Convergence
10.10 Alternating Series Error Bound
10.11 Finding Taylor Polynomial Approximations of Functions
10.12 Lagrange Error Bound
10.13 Radius and Interval of Convergence of Power Series
10.14 Finding Taylor or Maclaurin Series for a Function
10.15 Representing Functions as Power Series

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