Infinite sequence and seriesMade By:-

Enrolment no:-• 150860131044 • 150860131029• 150860131006• 150860131017• 150860131035• 150860131033• 150860131003• 150860131009• 150860131046• 150860131012

Subject code:-2110014

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

Infinite Sequence

Infinite Sequence An infinite sequence of numbers is a function from theset 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:

Bounded Sequence• The Bounded sequence is based on the

condition

There are two types of Bounded sequence:- Bounded above:- is said to be bounded above, if

there is some real number α such that Bounded Below:- is said to be bounded below,

is there exists a real number β such that

Sandwich Theorem or Squeeze Theorem• If for every and

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

Example: Study the following sequences and determine the type of sequenceSolution:

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

Infinite Series

Infinite Series

An infinite series is the sum of an infinite sequence of numbers:

+ + ….+ + ….

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:

nth Term Test

The Integral TestA positive term series +++………..+, Where decrease as n increase convergence & divergence as , it finite or infinite.

Direct Comparison TestComparison Test:

Limit Comparison Test

Ratio Test

Root Test

The Alternating Series TestTheorem: (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 0knc

Example: Test the convergence of Solution: Therefore, it is convergent.