Infinite series & sequence
- 1. Infinite sequence and series Made By:- Enrolment no:- • 150860131044 • 150860131029 • 150860131006 • 150860131017 • 150860131035 • 150860131033 • 150860131003 • 150860131009 • 150860131046 • 150860131012 Subject code:-2110014
- 2. Contents • Infinite sequence Bounded Sequence • Bounded above • Bounded below Sandwich Theorem or Squeeze Theorem Monotonic Sequence • Monotonically Increasing sequence • Monotonically decreasing sequence • Infinite series Zeroth Test Integral Test Comparison Test • Direct Comparison Test • Limit Comparison Test Ratio Test Root Test Alternating Series Test
- 4. Infinite Sequence An infinite sequence of numbers is a function from the set Z of integers into a set R. The set R can be any set, but in our course it is usually the set R of real numbers. Thus a sequence is usually denoted by writing down all the numbers in the range with numbers in the domain as the indices:
- 5. Bounded Sequence • The Bounded sequence is based on the condition {𝑎 𝑛} 𝑛≥1 There are two types of Bounded sequence:- Bounded above:- {𝑎 𝑛} 𝑛≥1 is said to be bounded above, if there is some real number α such that 𝑎 𝑛 ≤ α, ∀ 𝑛 Bounded Below:- {𝑎 𝑛} 𝑛≥1 is said to be bounded below, is there exists a real number β such that 𝑎 𝑛 ≥ β, ∀ 𝑛
- 7. Sandwich Theorem or Squeeze Theorem • If 𝑎 𝑛 ≤ 𝑏 𝑛 ≤ 𝑐 𝑛 for every 𝑛 ≥ 𝑛 𝑜 and
- 8. Monotone Sequences • We will begin with some terminology. • A sequence is called • Strictly increasing if • Increasing if • Strictly decreasing if • Decreasing if • A sequence that is either increasing or decreasing is said to be monotone, and a sequence that is either strictly increasing or strictly decreasing is said to be strictly monotone. 1nna ......321 naaaa ......321 naaaa ......321 naaaa ......321 naaaa
- 9. Example: Study the following sequences and determine the type of sequence Solution: 3, 4/3, 1, 6/7,... The sequence is decreasing. The inequality is satisfied for any value of n. The sequence is strictly monotonically decreasing
- 11. Infinite Series An infinite series is the sum of an infinite sequence of numbers: 𝑎1+ 𝑎2 + ….+ 𝑎 𝑛 + …. How can we find this sum of infinite numbers in a finite life time? For his, we look at the sequence of sums of finite number of terms, called the sequence of partial sums:
- 12. nth Term Test
- 13. The Integral Test A positive term series 𝑓1 +𝑓2 +𝑓3 +………..+𝑓𝑛 , Where 𝑓𝑛 decrease as n increase convergence & divergence as lim 𝑏→ ∞ 1 𝑏 𝑓 𝑥 𝑑𝑥, it finite or infinite.
- 15. Direct Comparison Test Comparison Test: Limit Comparison Test
- 17. Ratio Test
- 19. Root Test
- 20. The Alternating Series Test Theorem: (Alternating Series Test) Consider the series c1 - c2 + c3 - c4 . . . and -c1+ c2 - c3+ c4 . . . Where c1 > c2 > c3 > c4 > . . .> 0 and Then the series converge, and each sum S lies between any two successive partial sums. lim 0k n c Example: Test the convergence of 1 1 − 1 2 + 1 3 − 1 4 + ⋯ Solution: lim 𝑛→ ∞ (−1) 𝑛+1 1 𝑛 = lim 𝑛→ ∞ (−1) 𝑛+1 1 ∞ = 0 Therefore, it is convergent.