Porisma

Sequences and series

Learn one idea at a time

Choose a unit. Each has worked examples and practice.

  1. Read terms and check limits

    Distinguish terms from partial sums, describe behaviour and generate a recurrence.

  2. Work with arithmetic progressions

    Use a constant difference, relate terms and justify an arithmetic progression.

  3. Work with geometric progressions

    Use a common ratio, solve term conditions and check the first integer threshold.

  4. Connect terms, partial sums and sigma notation

    Recover a term, match summation bounds and distinguish a term from its running total.

  5. Change bounds and combine sums

    Count terms, remove prefixes and match a given summation identity.

  6. Sum an arithmetic progression

    Use pairing and finite sums, then check integer bounds in a model.

  7. Sum a finite geometric progression

    Choose a finite-sum formula and keep the first term and term count consistent.

  8. Check convergence before summing to infinity

    Use partial-sum limits and the ratio condition; distinguish a sequence limit from a series sum.

  9. Combine standard finite sums

    Use the natural-number, square and cube results while retaining every term.

  10. Model and analyse recurrences

    Generate terms, unfold a recurrence and separate candidate limits from justified behaviour.

  11. Model geometry, motion and repeated payments

    Define the term and observation time before combining progressions and sums.

  12. Review and choose practice

    Check the chapter skills and choose the next source question by method.

Read or print the complete chapter

By the end, you can

  • Distinguish a term from a sum
  • Use arithmetic, geometric and sigma formulas
  • Check convergence and analyse recurrences

12 focused units · about 2 hours for the full chapter. Allow extra time for practice.

The central idea

A sequence lists terms; a series adds them. For arithmetic terms, un=a+(n−1)du_n=a+(n-1)d and Sn=n(2a+(n−1)d)/2S_n=n(2a+(n-1)d)/2. For geometric terms, un=arn−1u_n=ar^{n-1} and Sn=a(1−rn)/(1−r)S_n=a(1-r^n)/(1-r) for r≠1r\ne1. An infinite geometric sum is a/(1−r)a/(1-r) only when ∣r∣<1|r|<1. Recurrences define each new term from earlier ones.

Terms, indices and partial sums

The index n in u1,u2,…,un,…u_1,u_2,\ldots,u_n,\ldots records position. A series adds terms, and Sn=u1+⋯+unS_n=u_1+\cdots+u_n is the sum of its first n terms. The teacher note stresses that n is a positive integer.

An explicit formula un=f(n)u_n=f(n) treats position as input and the term as output. For un=n2+3u_n=n^2+3, the sequence starts 4,7,12,19,28,…4,7,12,19,28,\ldots and u100=10003u_{100}=10003. Restrict a finite sequence’s indices to its listed positions.

Open the chapter exploration

Worked example

For 3,6,12,…3,6,12,\ldots, a=3a=3 and r=2r=2. The fourth term is 24, while the first four terms sum to 3(1−24)/(1−2)=453(1-2^4)/(1-2)=45. There is no finite sum to infinity.

A recurrence approaching a fixed point needs a stability argument; solving the fixed-point equation alone is not enough.

Check your understanding

For un=5(−1/2)n−1u_n=5(-1/2)^{n-1}, find the limiting term and the sum to infinity. Explain why these are different quantities. What happens to the partial sums if every term is 5 instead?

Practice questions

98 examples and practice questions, in 17 sections.

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