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Lecture 27 117 Power Series 118 Diﬀerentiation and Integration of Power Series Jiwen He 1 Power Series 11 Geometric Series and Variations Geometric Series. O I^EAH 7 ^ Z Z կ ¹ _ MU. Minimum element wrt K x is minimum element of S iﬀ for all λ ≻K∗ 0, x is the unique minimizer of λTz over S x S minimal element wrt K • if x minimizes λTz over S for some λ ≻K∗ 0, then x is minimal x1 S x2 λ1 λ2 • if x is a minimal element of a convex set S, then there exists a nonzero λ K∗ 0such that x minimizes.

1 are clear Also, if n< 1, then we have xn1 = 1 x n 1 = 1 x 1x n = 1 x 1 1 x n = xxn Hence x n1 = xx for all n2Z Now, let m2Z be arbitrary We will show that xmn= xmxnby. G is concave in t (for any choice of X ≻ 0, V);. ҋ y w p X ^ b t z W I i ԉ k j h ~ g ł̏h B H t i R H j B T x P i P ɂP 񂲖J A x I j B A ݕ B i R p ƁA ӗ 킸 Ȃ 炨 o v ܂ B j Ԓx ԃV t g Ȃ̂ŁA ߑO ɊC ɍs A ɂ݂̂ɍs 莞 Ԃ L Ɏg ܂ I I i ԂW F O O ` P W F O O/ x ԂP S F O O ` Q S F O O j.

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I A F Y Y 3 O X U Z F W O V U P N P D R B E E L N 7 L B A R N W A P 7 V 4 E I K Q V W Y X V Q U A Q V S P W R L Q Q B V Y T V H Q I 6 S V T P H Q A E N X K Z E E M P. ANXÀè åQýµ þY¶sŽMH·JÃ;1íš,Âèz €*T\UO·¨ÆbnzdÓŸR‰#ŸÒ‹1Ý @§b³ÿµ”t„ÿßTÆ×"PL ¸QÜ ÒåatiÐ)‘Ê’F® Fáœn Sƒ)èÀý HÇf“5b;Fe ÛùDÜ@=3éš†hš& U†U‡B( ÚQM¥ ¥¦ KšV % &Œ¤ƒ*{V Ý«ÚÉµ¹S÷Ö·©²Â“Æc eOéïJJçF é?# ¢¦»µ’Ò È«vaPVG³ ¤®¢Š( ¡¥ëWU½ÌV. M;n2Z De nition 73 tells us that xn1 = xx n(and hence x 1x 1 = xn) for all n2P First we show this is true for all n2Z The cases n= 0;.

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