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This document provides information about sequences and series in mathematics. It defines a sequence as an ordered set of real numbers denoted by a1, a2, a3, etc. where an represents the general term. Sequences can be finite or infinite. A sequence is said to converge to a limit l if the terms get arbitrarily close to l as n increases. Bounded sequences are those where there exists a number k such that an is always less than k. Monotonic sequences either always increase or decrease. A series is an infinite sum of terms of a sequence. The convergence, divergence or oscillation of a series depends on whether the partial sums sn tend to a finite limit, infinity, or fail to have a unique limit as
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Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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- 1. Mathematics Presentation Name: PRITAM MANDAL Department: Computer Science & Engineering Year: 2nd Semester: 3rd Roll no: 28100121048 Reg. no: 212810100110054
- 3. SEQUENCE S (1) An ordered set of real numbers is called a sequence and is denoted by If the number of terms is unlimited , then the sequence is said to be an infinite sequence and (an ) is its general term. For instance (i) 1,3,5,7,…,(2n-1),…, 3 a1,a2,a3,…...,an (an )
- 4. (i) 1,1/2,1/3,…,1/n,..., (ii) 1,-1,1,-1,…, (-1)(n-1) are infinite sequences. (2) Limit. A sequence is said to tend to a limit l , if for every ε >0 , a value N of n can be found such that |an-l| < for n ≥ N We then write or simply as 4 n→∞
- 5. (4) Bounded sequence: A sequence ( an ) is said to be bounded ,if there exists a number k such that for every n. (5) Monotonic sequence: the sequence ( an ) is said to increase steadily or to decrease steadily according as or for all values of n . 5 an+1≥an an+1≤an
- 6. “ Both increasing and decreasing sequences are called monotonic sequences. A monotonic sequence always tends to a limit , finite or infinite. Thus, a sequence which is monotonic and bounded is convergent. 6
- 7. SERIE S (1) Definition : If be an infinite sequence of real numbers , then Is called an infinite series. An infinite series is denoted by and the sum of its first n terms is denoted by . 7 u1,u2,u3,…...un u1+u2+u3+…...+un+..... ∞ sn
- 8. 8 (2) Convergence, Divergence and Oscillation of a series : Consider the infinite series And let the sum of the first n terms be Clearly , is a function of n and as n increases indefinitely three possibilities arise. sn
- 9. (i) If tends to a finite limit as , the series is said to be convergent. (ii) If sn tends to as ,the series is said to be divergent . (iii) If sn does not tend to a unique limit as , then the series is said to be oscillatory or non-convergent. sn
- 10. For your patience. Thank You! 10