James Stewart Quote
Notice that if , then and , whereas if, then and .(a) If is absolutely convergent, show that both of theseries and are convergent.(b) If is conditionally convergent, show that both of theseries and are divergent.44. Prove that if is a conditionally convergent series and is any real number, then there is a rearrangement of whose sum is . [Hints: Use the notation of Exercise 43. an an 0 an an 0an an 0 an 0 ananana nanana nanranrTake just enough positive terms so that their sum is greaterthan . Then add just enough negative terms so that thecumulative sum is less than . Continue in this manner and useTheorem 11.2.6.]45. Suppose the series is conditionally convergent.(a) Prove that the series is divergent.(b) Conditional convergence of is not enough to determine whether is convergent. Show this by giving anexample of a conditionally convergent series such thatconverges and an example where diverges.r anrann2anannannannananWe now have several ways of testing a series for convergence
Notice that if , then and , whereas if, then and .(a) If is absolutely convergent, show that both of theseries and are convergent.(b) If is conditionally convergent, show that both of theseries and are divergent.44. Prove that if is a conditionally convergent series and is any real number, then there is a rearrangement of whose sum is . [Hints: Use the notation of Exercise 43. an an 0 an an 0an an 0 an 0 ananana nanana nanranrTake just enough positive terms so that their sum is greaterthan . Then add just enough negative terms so that thecumulative sum is less than . Continue in this manner and useTheorem 11.2.6.]45. Suppose the series is conditionally convergent.(a) Prove that the series is divergent.(b) Conditional convergence of is not enough to determine whether is convergent. Show this by giving anexample of a conditionally convergent series such thatconverges and an example where diverges.r anrann2anannannannananWe now have several ways of testing a series for convergence
Related Quotes
Certainly we can say that the pace of modern life, increased and supported by our technology in general and our personal electronics in particular, has resulted in a short attention span and an addict...
About James Stewart
Born and raised in Indiana, Pennsylvania, Stewart started acting while at Princeton University. After graduating, he began a career as a stage actor making his Broadway debut in the play Carry Nation (1932). He landed his first supporting role in The Murder Man (1935) and had his breakthrough in Frank Capra's ensemble comedy You Can't Take It with You (1938). Stewart went on to receive the Academy Award for Best Actor for his performance in the George Cukor romantic comedy The Philadelphia Story (1940). His other Oscar-nominated roles were in Mr. Smith Goes to Washington (1939), It's a Wonderful Life (1946), Harvey (1950) and Anatomy of a Murder (1959).
Stewart played darker, more morally ambiguous characters in movies directed by Anthony Mann, including Winchester '73 (1950), The Glenn Miller Story (1954), and The Naked Spur (1953), and by Alfred Hitchcock in Rope (1948), Rear Window (1954), The Man Who Knew Too Much (1956), and Vertigo (1958). Stewart also starred in The Greatest Show on Earth (1952), The Spirit of St. Louis (1957), and The Flight of the Phoenix (1965) as well as the Western films How the West Was Won (1962), The Man Who Shot Liberty Valance (1962), and Cheyenne Autumn (1964).
With his private pilot's skills, he enlisted in the US Army Air Forces during World War II seeking combat duty and rose to be deputy commanding officer of the 2nd Bombardment Wing and commanding the 703d Bombardment Squadron from 1941 to 1947. He later transferred to the Air Force Reserve, and held various command positions until his retirement in 1968 as a brigadier general. Stewart remained unmarried until his 40s and was dubbed "The Great American Bachelor" by the press. In 1949, he married former model Gloria Hatrick McLean. They had twin daughters, and he adopted her two sons from her previous marriage. The marriage lasted until Gloria's death in 1994. Stewart died of a pulmonary embolism three years later.