Null Sequence and Dull Sequence in Mathematics

Null Sequence:

A sequence {s_n} is said to be null sequence if its limit is zero, that is,

.

In other words, a sequence that converges to a limit of zero is called the null sequence.

Example 1: The sequence {1/n} is a null sequence, because,

Example 2: The sequence {0} is a null sequence, because,

Dull Sequence:

A sequence {s_n} is said to be dull sequence if there is some N∈N such that for all .

Example: The sequence {s_n} = {} is a dull sequence, because for all .

Here, we notice that,

,

so this sequence is also null sequence. Hence, it is obvious that, every dull sequence is a null sequence.

Difference between Null Sequence and Dull Sequence:

The same terms are used in the definitions of both the null sequence and the dull sequence, but they are not in the same order, so the definitions of the null sequence and the dull sequence are not equivalent.

By definitions, every dull sequence is a null sequence but null sequences are not necessarily dull.

For example, The dull sequence {s_n}  = {} is a null sequence, but the null sequence {1/n} is not a dull sequence.

Null Sequence and Dull Sequence in Mathematics in Hindi

Null Sequence:

एक sequence (अनुक्रम) {s_n} को null sequence कहते है यदि इसकी सीमा (लिमिट) शून्य है, अर्थात,

.

दूसरे शब्दों में, एक अनुक्रम जो शून्य की सीमा में परिवर्तित (converges) होता है, null sequence कहलाता है।

Example 1: अनुक्रम {1/n} एक null sequence है, क्योंकि,

Example 2: अनुक्रम {0} एक null sequence है, क्योंकि,

Dull Sequence:

एक अनुक्रम {s_n} को Dull Sequence कहा जाता है यदि कुछ N∈N ऐसे हो कि सभी के लिए ।

उदाहरण:

अनुक्रम {s_n} = {} एक Dull Sequence है, क्योंकि सभी के लिए है।

यहाँ, हम देखते हैं कि,

,

तो यह अनुक्रम एक null sequence भी है। इसलिए, यह स्पष्ट है कि, प्रत्येक Dull Sequence एक null sequence है।

Null sequence और Dull Sequence के बीच अंतर:

Null sequence और Dull Sequence दोनों की परिभाषाओं में समान शब्दों का उपयोग किया जाता है, लेकिन वे एक ही क्रम में नहीं हैं, इसलिए Null sequence और Dull Sequence की परिभाषाएं समान नहीं हैं।.

परिभाषाओं के अनुसार, प्रत्येक Dull Sequence एक Null sequence है लेकिन Null sequence आवश्यक रूप से Dull Sequence नहीं हैं।

उदाहरण के लिए, Dull Sequence {s_n}  = {} एक Null sequence है, लेकिन Null sequence {1/n} एक Dull Sequence नहीं है।.

(Source – Various books from the college library)

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